Opera Medica et Physiologica - neuron
http://operamedphys.org/keywords/neuron
enModeling the Role of the Astrocyte Syncytium and K+ Buffering in Maintaining Neuronal Firing Patterns
http://operamedphys.org/content/modeling-role-astrocyte-syncytium-and-k-buffering-maintaining-neuronal-firing-patterns
<div class="field field-name-field-authors field-type-taxonomy-term-reference field-label-hidden clearfix"><ul class="links"><li class="taxonomy-term-reference-0"><a href="/authors/david-terman">David Terman</a> / <a href="/authors/min-zhou">Min Zhou</a></li></ul></div><div class="field field-name-field-author-affiliations field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="jqueryui-formatters accordion" data-instance="250-7-9"><h3><a href="#jqueryui-formatters-accordion">Author Affiliations</a></h3><div><p><p>David Terman<sup>1</sup> and Min Zhou<sup>2</sup></p>
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<p><sup>1</sup> Department of Mathematics, Ohio, <br /><sup>2</sup> Department of Neuroscience, Ohio State University.</p>
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</p></div></div></div></div></div><div class="field field-name-field-corresponding-author field-type-text-long field-label-inline clearfix"><div class="field-label">Corresponding author: </div><div class="field-items"><div class="field-item even"><p>David Terman (<a href="mailto:terman.1@osu.edu">terman.1@osu.edu</a>)</p>
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</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><h3 class="field-label">Keywords: </h3><ul class="links inline"><li class="taxonomy-term-reference-0" rel="dc:subject"><a href="/keywords/astrocyte" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Astrocyte</a></li><li class="taxonomy-term-reference-1" rel="dc:subject"><a href="/keywords/neuron" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">neuron</a></li><li class="taxonomy-term-reference-2" rel="dc:subject"><a href="/keywords/potassium-buffering" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">potassium buffering</a></li><li class="taxonomy-term-reference-3" rel="dc:subject"><a href="/keywords/gap-junctions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">gap junctions</a></li></ul></div><div class="field field-name-field-abstract field-type-text-long field-label-above"><div class="field-label">Abstract: </div><div class="field-items"><div class="field-item even"><p class="rtejustify">Computational models for two neuron/astrocyte networks are developed to explore mechanisms underlying the astrocytes’ role in maintaining neuronal firing patterns. For the first network, a single neuron receives periodic excitatory inputs at varying frequencies. We consider the role played by several astrocytic dendritic processes, including the Na+-K+ ATPase pump, K+ channels and gap junctions in maintaining extracellular ion homeostasis so that the neuron can faithfully sustain spiking in response to the excitatory input. The second network includes two neurons coupled through mutual inhibitory synapses. Here we consider the role of astrocytic dendritic processes in maintaining anti-phase or synchronous oscillations. Dynamical systems methods, including bifurcation theory and fast/slow analysis, is used to systematically reduce the complex model to a simpler set of equations. In particular, the first network, consisting of differential equations for the neuron and astrocyte membrane potentials, channel state variables and intracellular and extracellular Na+ and K+ concentrations, is reduced to a one dimensional map. Fixed points of the map determine whether the astrocyte can maintain extracellular K+ homeostasis so the neuron can respond to periodic input. </p>
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</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><div class="rtejustify"><strong>Introduction</strong></div>
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<div class="rtejustify">For many years, glial cells were considered to merely provide structural support, with no relevant contribution to cognitive functions. However, numerous studies over the past decade have demonstrated that glial cells are active players in signal processing and there is now little doubt that advanced brain function is achieved through dynamic interactions among neuronal and glia networks. Among glial cells, astrocytes are believed to perform many functions, including maintenance of extracellular ion balance, control of cerebral blood flow, modulation of synaptic transmission, and neurotransmitter uptake and release (Iadecola & Nedergaard, 2007; Barres, 2008; Newman, 2003; Giaume et al., 2010; Kouji & Newman, 2004; Attwell et al., 2010; Nedergaard et al., 2003). Several studies have demonstrated that astrocytes may play a critical role in the modulation of neuronal network activity, including oscillations and synchronization, mainly through their role in ion homeostasis and uptake of the neurotransmitters GABA and glutamate (Bellot-Saez et al., 2017; Poskanzer & Yuste, 2016; Szabo et al., 2017, Pannasch et al., 2011; Amiri et al., 2013; Chever et al., 2016; Haydon & Carmignoto, 2016). Finally, alterations in normal astrocytic physiology have been associated with several neuropsychiatric, neurodevelopmental, and neurodegenerative diseases, including epilepsy, stroke, Rett syndrome and Alzheimer’s disease (Allaman et al. 2011; Barres, 2008; Chen & Swanson, 2003; Kimelberg & Nedergaard, 2010; Nedergaard & Dirnagle, 2005; Somjen, 2001). </div>
<div class="rtejustify">An important role of astrocytes is to maintain a nearly constant extracellular K+ concentration in the face of neuronal activity that would tend to increase it (Kofuji & Newman, 2004; Bellot-Saez et al., 2007; Ransom 1996; Muller 1996). Failure of astrocytes to maintain proper extracellular K+ concentrations has been implicated in neurological diseases such as epilepsy and stroke (Dirnagl et al., 1999; Kimelberg & Nedergaard, 2010; Moskowitz et al., 2011; Nakase et al., 2003; Zhao & Rempe, 2010). A basic hypothesis for extracellular K+ clearance, or K+ spatial buffering, was introduced more than a half century ago (Kofuji & Newman, 2004; Orkand et al., 1966). However, recent experiments have demonstrated that astrocyte gap junction coupling may play a crucial role in extracellular ion homeostasis (Ma et al., 2016). In brief, strong gap junction coupling provides isopotentiality to the astrocytic network by minimizing the membrane potential depolarization that follows increased levels of K+; this, in turn, maintains a strong K+ inward driving force, which is critical for efficient astrocytic control of brain homeostasis. The syncytial isopotentiality also provides a stronger and sustained driving force for interastrocyte spatial transfer through gap junction channels and a sustained driving force for K+ release. </div>
<div class="rtejustify">In this study, we develop computational models for two neuron/astrocyte networks to explore mechanisms underlying the astrocytes’ role in maintaining neuronal firing patterns. For the first network, a single neuron receives periodic excitatory inputs at varying frequencies. We consider the role played by several astrocytic dendritic processes, including the Na+- K+ ATPase pump, K+ channels and gap junctions in maintaining extracellular ion homeostasis so that the neuron can faithfully sustain spiking in response to the excitatory input. The second network includes two neurons coupled through mutual inhibitory synapses. Here we consider the role of astrocytic dendritic processes in maintaining anti-phase or synchronous oscillations. </div>
<div class="rtejustify">A primary goal of this paper is to mathematically study the dependence of network behavior on various model parameters and cellular processes. This is done using dynamical systems methods, including bifurcation theory and fast/slow analysis, to systematically reduce the complex model to a simpler set of equations. In fact, we reduce the first network, consisting of differential equations for the neuron and astrocyte membrane potentials, channel state variables and intracellular and extracellular Na+ and K+ concentrations, to a one dimensional map. Fixed points of the map determine whether the astrocyte can maintain extracellular K+ homeostasis so the neuron can respond to periodic input. </div>
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<div class="rtejustify"><strong>Methods </strong></div>
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<div class="rtejustify">We consider two neuron/astrocyte network models. Each cell is modeled as a single compartment using the Hodgkin-Huxley formalism (Ermentrout & Terman, 2010). There are equations for the membrane potentials and ionic currents, as well as equations for Na+ and K+ concentrations both inside the cells and in the shared extracellular space. There are also equations for the neuronal and astrocytic Na+-K+ ATPase pumps. Simulations of the model were done using the numerical software XPPAUT (Ermentrout, 2002). </div>
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<div class="rtejustify"><strong><em>Network </em></strong></div>
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<div class="rtejustify">The first network consists of a single neuron that receives periodic excitatory input. The neuron is coupled to a single astrocyte through a shared extracellular space. The astrocyte is coupled to other astrocytes through gap junctions. </div>
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<div class="rtejustify"><em>Neuron</em>: The neuron’s membrane potential, VN, satisfies the equation</div>
<div class="rtejustify"><span class="math-tex">\(C_m \frac{dV_N}{dt}=-I_{Na}-I_K-I_{PN}-I_{exc}\)</span> (1)</div>
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<p>The first two terms on the right hand side of this equation correspond Na<sup>+</sup> and K<sup>+</sup> currents. These are defined as </p>
<p><span class="math-tex">\(I_{Na}=(g_{Na} m_∞^3 (V_N )h+g_{NaL} )(V_N-E_{Na} )\)</span></p>
<p><span class="math-tex">\(I_K=(g_k n^4+g_{KL} )(V_N-E_K )\)</span></p>
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<div class="rtejustify">
<p>Note that there are both Na<sup>+</sup> and K<sup>+</sup> leaks. As in (Huguet <em>et al</em>., 2016), we assume that <em>h = 1 − n</em> and n satisfies a differential equation of the form</p>
<p> <span class="math-tex">\(\frac{dn}{dt}=ϕ(n_∞ (V_N )-n)/τ_n (V_N)\)</span> (2)</p>
<p>If <em>X </em>= <em>m </em>or <em>n</em>, then</p>
<p> <span class="math-tex">\(X_∞ (V)=\frac{1}{1+e^{-(V-θ_X)/σ_x }}\)</span> and <span class="math-tex">\(τ_n (V)=τ_0+\frac{τ_1-τ_1}{1+e^{-(V-θ_n0)/σ_n0 }}\)</span> (3)</p>
<p>The Nernst potentials are given by</p>
<p><span class="math-tex">\(E_K=\frac{RT}{F} ln \frac{K_e}{K_i}\)</span> and <span class="math-tex">\(E_{Na}=\frac{RT}{F} ln \frac{Na_e}{Na_i} \)</span> (4)</p>
<p>where R,T and F are the gas constant, temperature and Faraday’s constant, respectively, and K<sub>i</sub>, K<sub>e</sub>, Na<sub>i</sub> and Na<sub>e</sub> are the K<sup>+</sup> and Na<sup>+</sup> concentrations in the neuron’s cytosol and extracellular space.</p>
<p>As in (Kager <em>et al</em>., 2002) the Na<sup>+</sup>-K<sup>+</sup> ATPase pump current is given by</p>
<p><span class="math-tex">\(I_{PN}=ρ_N (\frac{K_e}{2+K_e })^2 (\frac{Na_i}{7.7+Na_i} )^3\)</span> (5)</p>
<p>where <em>r</em><em><sub>N</sub></em> represents the maximal pump current.</p>
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<p>The term <em>I<sub>exc</sub></em>corresponds to periodic, excitatory synaptic input and is given by</p>
<p><span class="math-tex">\(I_{exc}=g_{exc} s(t)(V_N-E_{syn}) .\)</span></p>
<p>Suppose that the input is at <em>f<sub>r</sub></em> hz. For each integer <em>j</em> and <em>t<sub>j</sub> = j</em><em>*</em><em> f<sub>r</sub> /1000,</em> we let <em>s(t<sub>j</sub>) = 1</em> and</p>
<p><em>s’(t) = </em><em>-</em><em>b</em><em> s(t)</em> for <em>t<sub>j</sub> < t < t<sub>j+1.</sub></em></p>
<p>The neuron model parameters are: C<sub>m</sub> = 1 mF/cm<sup>2</sup>, g<sub>Na</sub> = 20, g<sub>K</sub> = 3, g<sub>NaL</sub> = .03, g<sub>KL</sub> = .2, j = .1, q<sub>m</sub> = -37, s<sub>m</sub> = 10, s<sub>n</sub> = 10, q<sub>n</sub> = -55, t<sub>0</sub> = .1, t<sub>1</sub> = 1, q<sub>n0</sub> = -40, s<sub>n0</sub> = -12, g<sub>exc</sub> = 2, b = 1 ms<sup>-1</sup>. Units for the maximal conductances (g’s) are mS/cm<sup>2</sup> and half activation variables (q’s) are mV.</p>
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<p><strong>Astrocyte:</strong> The astrocyte’s membrane potential, <em>V<sub>A</sub></em>, satisfies the equation</p>
<p><span class="math-tex">\(C_m^A \frac{dV_A}{dt}=-I_K^A-I_{Kir}-I_{Na}^A-I_{P}A-I_{gap}.\)</span> (6)</p>
<p>where <span class="math-tex">\(C_m^A\)</span>= 1u<span class="math-tex">\(\)</span>F/cm<sup>2</sup> and <span class="math-tex">\(I_K^A\)</span> and <span class="math-tex">\(I_Na^A\)</span> correspond to K<sup>+</sup> and Na<sup>+</sup> leak currents; these are given by</p>
<p><span class="math-tex">\(I_K^A=g_K^A (V_A- E_K^A) \)</span> and <span class="math-tex">\(I_{Na}^A=g_{Na}^A (V_A- E_{Na}^A)\)</span></p>
<p>The term <em>I<sub>Kir</sub></em> corresponds to an inward rectifying K<sup>+</sup> current (Ransom, 1996), and is given by</p>
<p><span class="math-tex">\(I_{Kir}=g_{Kir} \left({\frac{K_e^{1/2}}{1+e^{(V_A-E_K^A)/19.2}}}\right)(V_A-E_K^A)\)</span> (7)</p>
<p>The Nernst potentials, <span class="math-tex">\(E_K^A \)</span> and <span class="math-tex">\(E_{NA}^A\)</span>, as well as the ATPase pump current <em>I<sub>PA</sub></em><sub>,</sub> are defined very similar to (4) and (5).</p>
<p>The term <em>I<sub>gap</sub></em> corresponds to electrical coupling with another astrocyte that is electrically coupled with other astrocytes within a syncytium. We assume that this other astrocyte remains in steady state; that is, its membrane potential, , and intracellular K<sup>+</sup> and Na<sup>+</sup> concentrations, and , are constant. We assume that = -90 mV, = 135 mM and = 12 mM.</p>
<p>As in (Ma <em>et al</em>., 2016; Huguet <em>et al.</em>, 2016), we model <em>I<sub>gap</sub></em> using the Goldman-Hodgkin-Katz equation. That is, <em>I<sub>gap </sub>= I<sub>Kgap</sub> + I<sub>Nagap</sub></em> where</p>
<p><span class="math-tex">\(I_{Kgap}= P_{Kgap} F_{φA} \left(\frac{K_{iA} e^{(-φ_A )}-K_iA^0}{e^{-φ_A} -1} \right)\)</span> (8)</p>
<p><span class="math-tex">\(I_{Nagap}= P_{Nagap} Fφ_A \left( \frac{Na_{iA} e^{(-φ_A )}-Na_{iA}^0}{e^{-φ_A} -1} \right)\)</span></p>
<p>Here, <em><span class="math-tex">\(\varphi{A} = (F/RT)(V_A-V_A^0)\)</span>.</em> The constants <em>P<sub>Kgap</sub></em> and <em>P<sub>Nagap</sub></em> are the K<sup>+</sup> and Na<sup>+</sup> permeabilities, averaged over the entire cell membrane. Following (Ma <em>et al</em>., 2016), we assume that</p>
<p><strong><em>P<sub>Kgap</sub> = d<sub>gap</sub> P<sub>K</sub></em></strong> and <em> <strong> P<sub>Nagap </sub>= 0.8 P<sub>Kgap</sub></strong></em></p>
<p>where <em>P<sub>K</sub></em> = 6 *10<sup>−5 </sup>cm/s. We will vary <em>d<sub>gap</sub>, g<sup>A</sup><sub>K</sub></em> and <em>g<sub>Kir</sub></em> in the analysis to determine how the dynamics depends on gap junction coupling strength and K<sup>+</sup> currents.</p>
<p><strong>Ion concentrations: </strong>The neuron’s intracellular K<sup>+</sup> and Na<sup>+</sup> concentrations satisfy the equations</p>
<p><span class="math-tex">\(\frac{dK_i}{dt}=-\frac{10 S_N}{F Ω_N }(I_K-2I_{PN})\)</span> (9)</p>
<p><span class="math-tex">\(\frac{dNa_i}{dt}=-\frac{10 S_N}{F Ω_N }(I_{Na}-2I_{PN})\)</span></p>
<p>where <em>S<sub>N</sub></em> is the neuron’s surface area and W<sub>N</sub> is the volume of the neuron’s cytoplasm. The factor ’10’ is needed for consistency of units.</p>
<p>The astrocyte’s intracellular K<sup>+</sup> and Na<sup>+</sup> concentrations satisfy the equations</p>
<p><span class="math-tex">\(\frac{dK_iA}{dt}=-\frac{10 S_A}{F Ω_A }(I_K^A+I_{Kir}+I_{Kgap}-2I_{PA})\)</span></p>
<p><span class="math-tex">\(\frac{dNa_iA}{dt}=-\frac{10 S_A}{F Ω_A }(I_{Na}^A+I_{Kir}+I_{Nagap}-2I_{PA})\)</span></p>
<p>where <em>S<sub>A</sub></em> is the astrocyte’s surface area and Ω<sub>A</sub> is the volume of the astrocyte’s cytoplasm.</p>
<p>The extracellular K<sup>+</sup> and Na<sup>+</sup> ion concentrations satisfy the equations</p>
<p><span class="math-tex">\(\frac {dK_e}{dt}=\frac {10 S_N}{F Ω_N } (I_K-2I_PN )+\frac {10 S_A}{F Ω_A }(I_K^A+I_{Kir}+I_{Kgap}-2I_{PA})\)</span> (10)</p>
<p><span class="math-tex">\(\frac{dNa_e}{dt}=\frac{10 S_N}{F Ω_N} (I_{Na}+3I_PN )+\frac{10 S_A}{F Ω_A }(I_{Na}^A+I_{Nagap}+3I_PA)\)</span></p>
<p>where Ω<sub>E</sub> is the volume of the extracellular space, which is assumed to be a fixed fraction of the neuron’s volume; that is, Ω<sub>E</sub> = α<sub>0</sub> Ω<sub>N</sub><em>.</em> In the simulations, we let <em>S<sub>N</sub></em>= 10<sup>4</sup> μm<sup>2</sup>, Ω<sub>N</sub> = 5 10<sup>3</sup> μm<sup>3</sup>,<em> S<sub>A</sub></em> = 1.6 10<sup>3</sup> μm<sup>2</sup>, Ω<sub>A</sub> = 2 10<sup>3</sup> μm<sup>3</sup> and α<sub>0</sub> = .3.</p>
<p><strong>Network II</strong></p>
<p>Network II consists of two mutually coupled neurons. Each neuron shares extracellular space with an astrocyte, which, as before, is electrically coupled with another astrocyte. Both ’units’, consisting of a neuron, astrocyte and shared extracellular space, are modeled precisely as Network I above, except we replace the Na<sup>+</sup> and K<sup>+</sup> leaks in (1) by a single leak current, <em>I<sub>L</sub> = g<sub>L</sub>(V<sub>N </sub>− E<sub>L</sub>),</em> where <em>g<sub>L</sub></em> = .02 and <em>E<sub>L</sub></em> = −60 mV. Moreover, we replace the excitatory current, <em>I<sub>exc</sub>,</em> in (1) by a term corresponding to inhibitory synaptic input from the other neuron. That is, if j = 1 or 2, then we replace <em>I<sub>exc </sub></em>in the voltage equation for neuron j with</p>
<p><span class="math-tex">\(I_{syn}^j=g_{syn} s_k (V_N^j-E_{syn})\)</span></p>
<p>where <span class="math-tex">\(k \neq j \)</span> and <em>s<sub>k</sub></em> satisfies the equation </p>
<p><span class="math-tex">\(s_k^,=α(1-s_k ) s_∞ (V_N^k )-βs_k\)</span></p>
<p>Here, <em>E<sub>syn</sub></em> = −85 mV is the synaptic reversal potential and <em>s<sub>∞</sub></em> is defined as in (3). The parameter values for Network II are the same as Network I, except <em>θ<sub>n</sub></em>=−55, t<sub>1</sub> =2, <em>φ</em>=.2, <em>g<sub>syn</sub></em>=.05 and <em>β</em>=.18.</p>
<p><strong>Results</strong></p>
<p><em><strong>Solutions of the two networks</strong></em></p>
<p>Solutions of Network I are shown in Fig. 1. This figure demonstrates that whether the model can maintain repetitive spiking depends on several factors including the input frequency (<em>f<sub>r</sub>)</em>, the strengths of the Na<sup>+</sup>-K<sup>+</sup> ATPase pumps (<em>ρ<sub>N</sub></em>and <em>ρ<sub>A</sub></em>) and the strength of gap junction coupling (<em>d<sub>gap</sub></em>). Fig. 1A shows that the neuron can maintain a 30 hz firing rate for 5 seconds for moderate levels of gap junction coupling and Na<sup>+</sup>-K<sup>+</sup> ATPase pump strengths. However, as shown in Fig. 1B, if the input rate is increased to 40 hz, then the neuron stops firing at around 3 seconds, at which time it goes into so-called depolarization block. If we remove gap junction coupling by setting <em>d<sub>gap</sub> = 0</em>, then, as shown in Fig. 1C, the neuron cannot maintain periodic firing, even at a lower input rate of 10 hz. If we increase the Na<sup>+</sup>-K<sup>+</sup> ATPase pump strength, then, as shown in Fig. 1D, the neuron is able to maintain periodic firing.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure1_page-0001.jpg" target="_blank"><img alt="Figure 1: Solutions of Network I for different values of input frequency (fr), the strengths of the neuron’s Na+ -K+ ATPase pump (rN) and the strength of gap junction coupling (dgap). For each plot, rA = .5, �, b = 3 and gKir = 0. A) With gap junctions, the neuron can maintain firing for moderate pump strengths and input frequencies. B) Even with gap junctions, sufficiently high input rates lead to depolarization block. C) Without gap junctions, the neuron cannot maintain firing at 10 hz, unless, as shown in D), the pump strength is sufficiently high. " id="Figure 1: Solutions of Network I for different values of input frequency (fr), the strengths of the neuron’s Na+ -K+ ATPase pump (rN) and the strength of gap junction coupling (dgap). For each plot, rA = .5, �, b = 3 and gKir = 0. A) With gap junctions, the neuron can maintain firing for moderate pump strengths and input frequencies. B) Even with gap junctions, sufficiently high input rates lead to depolarization block. C) Without gap junctions, the neuron cannot maintain firing at 10 hz, unless, as shown in D), the pump strength is sufficiently high. " longdesc="Figure 1: Solutions of Network I for different values of input frequency (fr), the strengths of the neuron’s Na+ -K+ ATPase pump (rN) and the strength of gap junction coupling (dgap). For each plot, rA = .5, �, b = 3 and gKir = 0. A) With gap junctions, the neuron can maintain firing for moderate pump strengths and input frequencies. B) Even with gap junctions, sufficiently high input rates lead to depolarization block. C) Without gap junctions, the neuron cannot maintain firing at 10 hz, unless, as shown in D), the pump strength is sufficiently high. " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure1_page-0001.jpg" style="height:288px; width:500px" title="Figure 1: Solutions of Network I for different values of input frequency (fr), the strengths of the neuron’s Na+ -K+ ATPase pump (rN) and the strength of gap junction coupling (dgap). For each plot, rA = .5, �, b = 3 and gKir = 0. A) With gap junctions, the neuron can maintain firing for moderate pump strengths and input frequencies. B) Even with gap junctions, sufficiently high input rates lead to depolarization block. C) Without gap junctions, the neuron cannot maintain firing at 10 hz, unless, as shown in D), the pump strength is sufficiently high. " /></a></p>
<p><span style="font-size:11px"><em><strong>Figure 1:</strong> Solutions of Network I for different values of input frequency (fr), the strengths of the neuron’s Na+ -K+ ATPase pump (rN) and the strength of gap junction coupling (dgap). For each plot, rA = .5, g<sup>A</sup><sub>K</sub>, b = 3 and gKir = 0. A) With gap junctions, the neuron can maintain firing for moderate pump strengths and input frequencies. B) Even with gap junctions, sufficiently high input rates lead to depolarization block. C) Without gap junctions, the neuron cannot maintain firing at 10 hz, unless, as shown in D), the pump strength is sufficiently high. </em></span></p>
<p>In Fig. 2 we illustrate the range of values of the Na<sup>+</sup>-K<sup>+</sup> ATPase pump strengths (with <em>ρ<sub>N</sub> = ρ<sub>A</sub></em>) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. We consider the two cases: <em>(</em> <em>, g<sub>Kir</sub><sup>A</sup>) = (3, 0)</em> and <em>(</em> <em>, g<sub>Kir</sub>) = (0, 1). </em>We choose a smaller value for<em> g<sub>Kir</sub></em> to account for the K<sup>+</sup> dependence of <em>I<sub>Kir </sub></em> open probability. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve.</p>
<p>Without gap junctions (<em>d<sub>gap</sub> = 0</em>), there is almost no difference between these two cases in the frequencies at which the neuron can maintain steady spiking. In particular, without gap junctions, the inward rectifying K<sup>+</sup> current does not seem to enhance K<sup>+</sup> buffering, which is needed to prevent <em>K<sub>e</sub></em> from rising above the threshold for depolarization block.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure2_page-0001.jpg" target="_blank"><img alt="Figure 2: The range of values of the Na+-K+ ATPase pump strengths (with N = A) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve. The dgap = 0 (blue) curve corresponds to both cases: (g_K^A, gKir) = (0,1) and (g_K^A, gKir)= (3,0). " id="Figure 2: The range of values of the Na+-K+ ATPase pump strengths (with N = A) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve. The dgap = 0 (blue) curve corresponds to both cases: (g_K^A, gKir) = (0,1) and (g_K^A, gKir)= (3,0). " longdesc="Figure 2: The range of values of the Na+-K+ ATPase pump strengths (with N = A) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve. The dgap = 0 (blue) curve corresponds to both cases: (g_K^A, gKir) = (0,1) and (g_K^A, gKir)= (3,0). " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure2_page-0001.jpg" style="height:288px; width:500px" title="Figure 2: The range of values of the Na+-K+ ATPase pump strengths (with N = A) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve. The dgap = 0 (blue) curve corresponds to both cases: (g_K^A, gKir) = (0,1) and (g_K^A, gKir)= (3,0). " /></a></p>
<p><span style="font-size:11px"><em><strong>Figure 2: </strong>The range of values of the Na<sup>+</sup>-K<sup>+</sup> ATPase pump strengths (with r<sub>N</sub> = r<sub>A</sub>) and input frequencies for which the neuron is able to maintain spiking for 10 seconds. The neuron exhibits depolarization block for values of pump strengths and input frequencies above each curve. The d<sub>gap</sub> = 0 (blue) curve corresponds to both cases: ( , g<sub>Kir</sub>) = (0,1) and ( , g<sub>Kir</sub>)= (3,0).</em></span></p>
<p>With gap junctions (<em>d<sub>gap</sub> = 1</em>), the neuron can maintain higher firing rates with just <em>I<sub>Kir </sub> </em>than with just if the pump strengths are sufficiently strong. This is mainly because the threshold for <em>K<sub>e </sub></em>when the neuron exhibits depolarization block increases at higher pump strengths. (This will be demonstrated later.) Large <em>K<sub>e</sub></em> values strengthen <em>I<sub>Kir</sub></em> and enhance K<sup>+</sup> buffering.</p>
<p>Fig. 3 shows solutions of Network II. With gap junction coupling (<em>d<sub>gap</sub> = 1</em>), the network maintains anti-phase spiking. However, without gap junction coupling (<em>d<sub>gap</sub> = 0</em>) the network switches from anti-phase to synchronous spiking at around 4.5 seconds. For this simulation, <em>g<sub>Kir </sub>= 1</em> and <em>= 0</em>. The result is almost identical if instead, <em>g<sup>A</sup><sub>Kir</sub> = 0</em> and g<sup>A</sup><sub>K</sub><em> = 3.</em></p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure3_page-0001.jpg" target="_blank"><img alt="Figure 3: Solutions of Network II. A) With gap junctions, the two neurons exhibit anti-phase oscillations. B) Without gap junction coupling, the network switches from anti-phase to synchronous spiking at around 4.5 seconds. C) With gap junctions, the model maintains a nearly constant Ke level, but not without gap junctions. D) Blow up of solution shown in B). " id="Figure 3: Solutions of Network II. A) With gap junctions, the two neurons exhibit anti-phase oscillations. B) Without gap junction coupling, the network switches from anti-phase to synchronous spiking at around 4.5 seconds. C) With gap junctions, the model maintains a nearly constant Ke level, but not without gap junctions. D) Blow up of solution shown in B). " longdesc="Figure 3: Solutions of Network II. A) With gap junctions, the two neurons exhibit anti-phase oscillations. B) Without gap junction coupling, the network switches from anti-phase to synchronous spiking at around 4.5 seconds. C) With gap junctions, the model maintains a nearly constant Ke level, but not without gap junctions. D) Blow up of solution shown in B). " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure3_page-0001.jpg" style="height:308px; width:500px" title="Figure 3: Solutions of Network II. A) With gap junctions, the two neurons exhibit anti-phase oscillations. B) Without gap junction coupling, the network switches from anti-phase to synchronous spiking at around 4.5 seconds. C) With gap junctions, the model maintains a nearly constant Ke level, but not without gap junctions. D) Blow up of solution shown in B). " /></a></p>
<p><em><span style="font-size:11px"><strong>Figure 3:</strong> Solutions of Network II. A) With gap junctions, the two neurons exhibit anti-phase oscillations. B) Without gap junction coupling, the network switches from anti-phase to synchronous spiking at around 4.5 seconds. C) With gap junctions, the model maintains a nearly constant K<sub>e</sub> level, but not without gap junctions. D) Blow up of solution shown in B).</span></em></p>
<p><em><strong>Isopotentiality and K<sup>+</sup> buffering</strong></em></p>
<p>Our results demonstrate that astrocytic gap junctional coupling plays a critical role in maintaining neuronal firing patterns. The basic mechanism underlying this behavior is so-called <em>K<sup>+</sup> spatial buffering:</em> gap junction coupling allows astrocytes to maintain a nearly constant extracellular K<sup>+</sup> concentration in the face of neuronal activity that would tend to increase it (Kofuji & Newman, 2004; Orkand <em>et al</em>., 2006). If the gap junction coupling strength is weak, then the astrocytes are not able to clear elevated extracellular K<sup>+</sup> levels. This leads to increased neuronal excitability, which may change their firing properties. When extracellular K<sup>+</sup> concentrations rise above some threshold, the neurons exhibit depolarization block. </p>
<p>How does strong gap junction coupling lead to K<sup>+</sup> spatial buffering? This depends on so-called isopotentiality; that is, an astrocyte’s associated syncytium provides powerful electrical coupling that equalizes the astrocyte’s membrane potential with its neighbors (Ma <em>et al</em>., 2016). To understand why isopotentiality leads to K<sup>+</sup> spatial buffering, we first note that K<sup>+</sup> currents are normally outward; that is, they allow for the flow of K<sup>+</sup> ions from the cell’s inside to the extracellular space. In general, we can express a K<sup>+</sup> current as <em>I<sub>K</sub> = g<sub>K</sub>(V<sub>A</sub> − </em> <em>).</em> If <em>V<sub>A</sub> > </em> <em>,</em> then <em>I<sub>K</sub> > 0</em> and the current is outward. In order for the K<sup>+</sup> currents to be inward, the cell’s membrane potential must lie below the K<sup>+</sup> reversal potential. When neurons spike, they release K<sup>+</sup> into the extracellular space. Hence, <em>K<sub>e</sub></em> increases, as does = (RT/F) ln(<em>K<sub>e</sub>/K<sub>iA</sub></em>). However, because of isopotentiality, the astrocyte’s membrane potential remains nearly constant. This allows for <em>V<sub>A</sub> <</em> , so that the astrocyte’s K<sup>+</sup> currents become inward and are, therefore, able to clear K<sup>+</sup> from the extracellular space.</p>
<p>In Fig. 4, we consider Network I and plot the response of <em>V<sub>A</sub></em> and <em>K<sub>e</sub></em> to different excitatory input rates with (Fig. 4A,B) and without (Fig. 4C,D) gap junction coupling. When <em>d<sub>gap</sub> = 1</em>, both the astrocyte’s membrane potential and extracellular K<sup>+</sup> concentration remain nearly constant. However, when there are no gap junctions (<em>d<sub>gap</sub> = 0</em>), there is a steady rise in both <em>V<sub>A</sub></em>and <em>K<sub>e</sub></em> even at low input rates.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure4_page-0001.jpg" target="_blank"><img alt="Figure 4: Solutions of Network I showing VA and Ke. Here, gKir = 3 and g_K^A = 0. A,B) With gap junctions the network can maintain nearly constant VA and Ke , even at 20 hz input. C,D) This is not the case if there are no gap junctions. " id="Figure 4: Solutions of Network I showing VA and Ke. Here, gKir = 3 and g_K^A = 0. A,B) With gap junctions the network can maintain nearly constant VA and Ke , even at 20 hz input. C,D) This is not the case if there are no gap junctions. " longdesc="Figure 4: Solutions of Network I showing VA and Ke. Here, gKir = 3 and g_K^A = 0. A,B) With gap junctions the network can maintain nearly constant VA and Ke , even at 20 hz input. C,D) This is not the case if there are no gap junctions. " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure4_page-0001.jpg" style="height:288px; width:500px" title="Figure 4: Solutions of Network I showing VA and Ke. Here, gKir = 3 and g_K^A = 0. A,B) With gap junctions the network can maintain nearly constant VA and Ke , even at 20 hz input. C,D) This is not the case if there are no gap junctions. " /></a></p>
<p><em><span style="font-size:11px"><strong>Figure 4:</strong> Solutions of Network I showing V<sub>A</sub> and K<sub>e</sub>. Here, g<sub>Kir</sub> = 3 and = 0. A,B) With gap junctions the network can maintain nearly constant V<sub>A</sub> and K<sub>e</sub> , even at 20 hz input. C,D) This is not the case if there are no gap junctions.</span></em></p>
<p>Fig. 5 shows plots of <em>V<sub>A</sub> ,</em> and <em>I<sub>Kir</sub></em> , with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, <em>V<sub>A</sub></em> falls below and <em>I<sub>Kir</sub></em> reverses (<em>I<sub>Kir</sub></em>< 0). Without gap junctions, <em>V<sub>A</sub></em> tracks very closely with and <em>I<sub>Kir</sub></em>remains negligible.</p>
<p>Fig. 3C shows that for Network II, with strong gap junction coupling, <em>K<sub>e</sub></em> remains nearly constant; however, without gap junction coupling, there is a steady rise in <em>K<sub>e</sub></em> during neuronal firing.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure5_page-0001.jpg" target="_blank"><img alt="Figure 5: Plots of A) VA and E_K^A, and B) IKir, with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, VA falls below E_K^A and IKir reverses (IKir < 0). Without gap junctions, VA tracks very closely with E_K^A and IKir remains negligible. " id="Figure 5: Plots of A) VA and E_K^A, and B) IKir, with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, VA falls below E_K^A and IKir reverses (IKir < 0). Without gap junctions, VA tracks very closely with E_K^A and IKir remains negligible. " longdesc="Figure 5: Plots of A) VA and E_K^A, and B) IKir, with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, VA falls below E_K^A and IKir reverses (IKir < 0). Without gap junctions, VA tracks very closely with E_K^A and IKir remains negligible. " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure5_page-0001.jpg" style="height:215px; width:500px" title="Figure 5: Plots of A) VA and E_K^A, and B) IKir, with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, VA falls below E_K^A and IKir reverses (IKir < 0). Without gap junctions, VA tracks very closely with E_K^A and IKir remains negligible. " /></a></p>
<p><em><span style="font-size:11px"><strong>Figure 5:</strong> Plots of A) V<sub>A</sub> and , and B) I<sub>Kir</sub>, with and without gap junctions, for the same solutions shown in Fig. 4 with an input rate of 10 hz. With gap junctions, V<sub>A</sub> falls below and I<sub>Kir</sub> reverses (I<sub>Kir</sub> < 0). Without gap junctions, V<sub>A</sub> tracks very closely with and I<sub>Kir</sub> remains negligible.</span></em></p>
<p><em><strong>Analysis</strong></em></p>
<p> </p>
<p>We mathematically analyze Network I by first making some simplifying assumptions and then reducing the full model to a simpler set of equations. The analysis leads to a single equation for just <em>K<sub>e</sub></em>. This will be used to determine the <em>K<sub>e</sub></em> threshold for when the neuron exhibits depolarization block and help explain the response of the model to excitatory input.</p>
<p>We begin by noting that the total amounts of K<sup>+</sup> and Na<sup>+</sup> ions are conserved. That is </p>
<p><span class="math-tex">\(Ω_e K_e+Ω_N K_i+Ω_A K_iA=K_{tot}\)</span> (11)</p>
<p>and</p>
<p><span class="math-tex">\(Ω_e {Na}_e+Ω_N Na_i+Ω_A Na_iA=Na_{tot}\)</span></p>
<p><span style="font-family:arial,sans-serif; font-size:11.0pt">are constant. We assume that </span></p>
<p><span class="math-tex">\(K_{tot}=Ω_e K_e^0+Ω_N K_i^0+Ω_A K_iA^0\)</span> and <span class="math-tex">\(Na_{tot}=Ω_e Na_e^0+Ω_N Na_i^0+Ω_A Na_iA^0\)</span> </p>
<p>where</p>
<p><span class="math-tex">\(K_e^0=4 \)</span> mM, <span class="math-tex">\(Na_e^0\)</span> =135 mM, <span class="math-tex">\(K_i^0=K_iA^0=\)</span>135 mM and <span class="math-tex">\(Na_i^0=Na_iA^0=\)</span>12 mM.</p>
<p>We now make several assumptions in order to derive a reduced model. We first assume intracellular neuron electroneutrality; that is, <em>K<sub>i</sub> + Na<sub>i</sub></em> = + is constant. Our next assumption is that the astrocyte’s intracellular K<sup>+</sup> and Na<sup>+</sup> concentrations are constant; that is, and . With these assumptions we can solve for all of the ion concentrations in terms of <em>K<sub>e</sub></em> and the Network I model reduces to equations for just <em>V<sub>N</sub> , n, V<sub>A</sub></em> and <em>K<sub>e</sub></em>.</p>
<p>We can reduce the model further using fast/slow analysis. The membrane potential <em>V<sub>A</sub></em> clearly evolves on a time-scale much faster than the ion concentration <em>K<sub>e</sub></em>. Since the astrocyte does not ’spike’, we may assume that <em>V<sub>A</sub></em> is close to steady state; that is, the right hand side of (6) is, to leading order, zero and we can solve for <em>V<sub>A</sub></em> in terms of the other variables. Note that if there is no<em> I<sub>Kir</sub></em> current (<em>g<sub>Kir</sub> = </em>0) then the right hand side of (6) is, in fact, a linear function of <em>V<sub>A</sub></em>. This is because if <span class="math-tex">\(K_{iA}=K_{iA}^0\)</span> and <span class="math-tex">\(Na_{iA}=Na_{iA}^0\)</span> , then the gap junction current can be written as</p>
<p><span class="math-tex">\(I_{gap}=g_{gap} (V_A-V_A^0)\)</span></p>
<p>where</p>
<p><span class="math-tex">\(g_{gap}=\frac{F^2}{RT} d_{gap} P_K (K_{iA}^0+.8Na_{iA}^0 ).\)</span> (12)</p>
<p><span style="font-family:arial,sans-serif; font-size:11.0pt">Setting the right hand side of (6) equal to zero, we find that</span></p>
<p><span style="font-family:arial,sans-serif; font-size:11.0pt"><span class="math-tex">\(V_A=\frac{g_K^A E_K^A+g_{Na}^A E_{Na}^A+I_{PA}-I_{exc}}{g_K^A+g_{Na}^A-g_{gap}}\)</span></span> (13)</p>
<p>The full Network I model is now reduced to equations for just <em>V<sub>N</sub>, n</em> and <em>K<sub>e</sub></em>, which we write as</p>
<p><span class="math-tex">\(C_m \frac{dV_N}{dt}=-I_{Na}-I_K-I_{PN}-I_{exc}\)</span></p>
<p><span class="math-tex">\(\frac{dn}{dt}=\frac{ϕ(n_∞ (V_N )-n)}{τ_n (V_N )}\)</span></p>
<p><span class="math-tex">\(\frac{dK_e}{dt}=Φ(V_N,n,K_e )\)</span> (14)</p>
<p>Assume, for now, that there is no excitatory input. Then fast/slow analysis is used to analyze the reduced system. If we consider the slow variable <em>K<sub>e</sub></em> to be a bifurcation parameter in the fast subsystem for <em>(V<sub>N</sub>, n)</em>, then the resulting bifurcation diagram is shown in Fig. 6. Note that there is a stable fixed point for <span class="math-tex">\(K_e<K_HB^1≈10.35\)</span> and <span class="math-tex">\(K_e>K_{HB}^2≈25.75\)</span>. There is a subcritical Hopf bifurcation at and a supercritical Hopf bifurcation at . Moreover, there are stable periodic orbits for <span class="math-tex">\(K_{HB}^1<K_e<K_{HB}^2.\)</span></p>
<p>We next consider the evolution of the slow variable <em>K<sub>e</sub></em>. This is done using the method of averaging. Denote the fixed points of the fast subsystem as <em>(V<sub>fp</sub>(K<sub>e</sub>), n<sub>fp</sub>(K<sub>e</sub>))</em> and the stable periodic orbits as <em>(V<sub>P</sub>(t, K<sub>e</sub>), n<sub>P</sub>(t, K<sub>e</sub>))</em>. Let <em>T(K<sub>e</sub>) </em>be the period of the periodic orbits. Near the branch of stable fixed points, <em>K<sub>e</sub></em> satisfies, to leading order,</p>
<p><span class="math-tex">\(\frac{dK_e}{dt}= Φ(V_{fp} (K_e ),n_{fp} (K_e ),K_e ) =Φ_{fp} (K_e )\)</span></p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure6_page-0001.jpg" target="_blank"><img alt="Figure 6: Bifurcation diagram for (14) with bifurcation parameter Ke. There are both supercritical and subcritical Hopf bifurcations and a branch of stable periodic orbits. " id="Figure 6: Bifurcation diagram for (14) with bifurcation parameter Ke. There are both supercritical and subcritical Hopf bifurcations and a branch of stable periodic orbits. " longdesc="Figure 6: Bifurcation diagram for (14) with bifurcation parameter Ke. There are both supercritical and subcritical Hopf bifurcations and a branch of stable periodic orbits. " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure6_page-0001.jpg" style="height:288px; width:500px" title="Figure 6: Bifurcation diagram for (14) with bifurcation parameter Ke. There are both supercritical and subcritical Hopf bifurcations and a branch of stable periodic orbits. " /></a></p>
<p><span style="font-size:11px"><em><strong>Figure 6:</strong> Bifurcation diagram for (14) with bifurcation parameter K<sub>e</sub>. There are both supercritical and subcritical Hopf bifurcations and a branch of stable periodic orbits.</em></span></p>
<p> </p>
<p>For <span class="math-tex">\(K_{HB}^1<K_e<K_{HB}^2, K_e\)</span> satisfies, to leading order,</p>
<p><span class="math-tex">\(\frac{dK_e}{dt} = \frac{1}{T(K_e)} ∫_0^{T(K_e)}Φ(V_P (t,K_e ),n_P (t,K_e ),K_e ) dt = Φ_{ave} (K_e )\)</span></p>
<p>We compute <span class="math-tex">\(Φ_{fp} (K_e )\)</span> and <span class="math-tex">\(Φ_{ave} (K_e )\)</span> numerically and the result is shown in Fig. 7A.</p>
<p>We first consider the <em>K<sub>e</sub></em> dynamics near the branch of stable fixed points of the fast subsystem with . Note that there exists <em>K<sub>P</sub></em> < <em>K<sup>1</sup></em><sub><em>HB</em></sub> such that <span class="math-tex">\(Φ_{fp} (K_e )=0\)</span> . Moreover, <span class="math-tex">\( Φ_{fp} (K_e )>0\)</span> for <em>0 < K<sub>e</sub> < K<sub>P</sub></em> and <span class="math-tex">\(Φ_{fp} (K_e )<0\)</span>for <em>K<sub>P</sub> < K<sub>e</sub> <</em> <em>K<sup>1</sup></em><sub><em>HB</em></sub>. This implies that <em>K<sub>e</sub> → K<sub>P</sub></em> as <em>t → ∞.</em></p>
<p> </p>
<p>We next consider the spiking regime when . As shown in Fig. 7A, . Hence, <em>K<sub>e</sub></em> must increase and the neuron must approach depolarization block.</p>
<p>This analysis demonstrates that is the threshold for when the neuron exhibits depolarization block. Fig. 7 shows that this threshold is an increasing function of the neuron’s Na<sup>+</sup>-K<sup>+</sup> ATPase pump strength.</p>
<p>We next consider the neuron’s response to periodic excitatory input. Whenever the neuron spikes, there is also a spike in <em>I<sub>K</sub></em>. This leads to a fast rise in <em>K<sub>e</sub></em>. We now assume that at the spike times, <em>K<sub>e</sub></em>increases by a fixed amount, which we denote as <em>K</em><em><sub>d</sub></em>. Based on numerics (see Fig. 4), we let <em>K</em><em><sub>d</sub></em> = .38 mM.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure7_page-0001.jpg" target="_blank"><img alt="Figure 7: A) Plots of fp(Ke) and ave(Ke). The threshold for depolarization block is Ke = K_HB^1. B) Plot of the depolarization block threshold versus the neuron pump strength N." id="Figure 7: A) Plots of fp(Ke) and ave(Ke). The threshold for depolarization block is Ke = K_HB^1. B) Plot of the depolarization block threshold versus the neuron pump strength N." longdesc="Figure 7: A) Plots of fp(Ke) and ave(Ke). The threshold for depolarization block is Ke = K_HB^1. B) Plot of the depolarization block threshold versus the neuron pump strength N." src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure7_page-0001.jpg" style="height:213px; width:500px" title="Figure 7: A) Plots of fp(Ke) and ave(Ke). The threshold for depolarization block is Ke = K_HB^1. B) Plot of the depolarization block threshold versus the neuron pump strength N." /></a></p>
<p><strong>Figure 7: </strong>A) Plots of Ф<sub>fp</sub>(<em>K<sub>e</sub></em>) and Ф<sub>ave</sub>(<em>K<sub>e</sub></em>). The threshold for depolarization block is <em>K<sub>e</sub></em> =K<sup>1</sup><sub>HB</sub> . B) Plot of the depolarization block threshold versus the neuron pump strength <em>r</em><em><sub>N</sub></em>.</p>
<p>Between spike times, <em>K<sub>e</sub></em> satisfies (14), which depends on <em>V<sub>N </sub></em> and <em>n</em>. We use fast/slow analysis to express these other variables in terms of <em>K<sub>e</sub></em>. This is done by first setting the right hand sides of (1) and (2) equal to zero and then solving for <em>(V<sub>N</sub>, n)</em> in terms of <em>K<sub>e</sub></em>. However, the right hand side of (1) is a nonlinear function of <em>V<sub>N</sub></em> and <em>n</em>. In order to obtain an explicit formula for <em>V<sub>N</sub></em>, we note that between spikes, <em>V<sub>N</sub></em> is near a resting state and the channel activation terms, <span class="math-tex">\(m_∞^3 (V_N) \)</span> and <em>n<sup>4</sup></em>, are very small. We, therefore, use the approximation </p>
<p><strong><em>I<sub>Na</sub> ≈ g<sub>NaL</sub>(V<sub>N</sub> − E<sub>Na</sub>)</em></strong> and <strong><em>I<sub>K</sub> ≈ g<sub>KL</sub>(V<sub>N</sub> − E<sub>K</sub>)</em>.</strong> (15)</p>
<p>In this case, the right hand side of (1) is linear in <em>V<sub>N</sub></em> and does not depend on <em>n</em>. Setting the right hand side of (1) equal to zero, we find that</p>
<p><span class="math-tex">\(V_N=\frac{g_{NaL} E_{Na}+g_{KL} E_K-I_PN}{g_{NaL}+g_{KL} }\)</span> (16)</p>
<p>We can now express F in (14) as a function of just <em>K<sub>e</sub></em>; we denote this function as <span class="math-tex">\(\psi(Ke)\)</span><em>.</em> This is done by first replacing <em>I<sub>K</sub></em> by the approximation given in (15) and then using (16).</p>
<p>In summary, the reduced model is the following. Suppose that the excitatory input is at <em>f<sub>r</sub></em> hz. For each integer<em> j</em> and <em>t<sub>j</sub> = j · f<sub>r </sub></em>/1000,</p>
<p><span class="math-tex">\(K_e (t_j^+ )=K_e (t_j^- )+K_δ\)</span></p>
<p>and</p>
<p><span class="math-tex">\(\frac{dK_e}{dt}=Ψ(K_e ) \)</span> for <em>t<sub>j</sub> < t < t<sub>j+1</sub></em> (17)</p>
<p>Solutions of this reduced model are shown in Fig. 8. With gap junctions (Fig. 8A) the neuron can maintain firing at 10 and 20 hz, since, in both cases, <em>K<sub>e</sub></em>remains below the threshold for depolarization block. Without gap junctions (Fig. 8B) the neuron cannot maintain firing at 10 hz if <em>ρ<sub>N</sub></em> = .5, since <em>K<sub>e</sub></em> increases past this threshold. However, increasing the neuron’s Na<sup>+</sup>-K<sup>+</sup> ATPase pump strength to <em>ρ<sub>N</sub></em> = 1 allows the neuron to maintain a 10 hz firing rate.</p>
<p>Finally, we construct a 1-dimensional map; fixed points of the map correspond to the asymptotic behavior of <em>K<sub>e</sub></em> for the reduced model. To define the map, fix <em>K<sub>0</sub></em> > 0 and let <em>K<sub>e</sub>(t; K<sub>0</sub>)</em> be the solution of (17) with <em>K<sub>e</sub>(0; K<sub>0</sub>) = K<sub>0</sub></em>. Then the map is defined as simply</p>
<p><span class="math-tex">\(Π(K_0 )=K_e (\frac{f_r}{1000};K_0 )+K_δ\)</span></p>
<p>Note that a fixed point of this map corresponds to a periodic solution of the reduced model (17).</p>
<p>In Fig. 8C,D we plot P(<em>K<sub>e</sub></em>)−<em>K<sub>e</sub></em> for the same parameter values used in Fig. 8A,B. Fixed points of P(<em>K<sub>e</sub></em>) correspond to zeros of the corresponding curves. Note in Fig. 8D that if <em>d<sub>gap</sub></em> =0 and <em>ρ<sub>N</sub></em> =.5, then P(<em>K<sub>e</sub></em>) > <em>K<sub>e</sub></em>for all values of <em>K<sub>e</sub></em> below the threshold for depolarization block. Hence, <em>K<sub>e</sub></em> must steadily increase past the threshold.</p>
<p><a href="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure8_page-0001.jpg" target="_blank"><img alt="Figure 8: A,B) Solutions of the reduced model (17). The dashed line corresponds to the threshold for depolarization block. C,D) Plots of the map (Ke) − Ke. Fixed points (Ke) correspond to zeros of the corresponding curves. " id="Figure 8: A,B) Solutions of the reduced model (17). The dashed line corresponds to the threshold for depolarization block. C,D) Plots of the map (Ke) − Ke. Fixed points (Ke) correspond to zeros of the corresponding curves. " longdesc="Figure 8: A,B) Solutions of the reduced model (17). The dashed line corresponds to the threshold for depolarization block. C,D) Plots of the map (Ke) − Ke. Fixed points (Ke) correspond to zeros of the corresponding curves. " src="http://www.operamedphys.org/sites/default/files/pictures/pictures/articles/2019_ISSUE_1_MARCH/terman/figure8_page-0001.jpg" style="height:395px; width:500px" title="Figure 8: A,B) Solutions of the reduced model (17). The dashed line corresponds to the threshold for depolarization block. C,D) Plots of the map (Ke) − Ke. Fixed points (Ke) correspond to zeros of the corresponding curves. " /></a></p>
<p><span style="font-size:11px"><em><strong>Figure 8:</strong> A,B) Solutions of the reduced model (17). The dashed line corresponds to the threshold for depolarization block. C,D) Plots of the map P(K<sub>e</sub>) − K<sub>e</sub>. Fixed points P(K<sub>e</sub>) correspond to zeros of the corresponding curves.</em></span></p>
<p> </p>
<p><strong>Discussion</strong></p>
<p>The main goals of this paper were to: 1) develop computational models to study mechanisms underlying astrocytes’ role in maintaining neuronal firing patterns; and 2) use mathematical tools to systematically reduce the complex model to a simpler system in order to characterize how solutions depend on network parameters and cellular processes. Simulations of the computational model demonstrate the importance of gap junctional coupling in K<sup>+</sup> spatial buffering and, thereby preventing elevated levels of extracellular K<sup>+</sup> leading to depolarization block. Using dynamical systems methods, we reduced the full Network I model to a one-dimensional map. Fixed points of the map determine whether the astrocyte can maintain extracellular K<sup>+</sup> homeostasis so the neuron can faithfully respond to periodic input.</p>
<p>The basic mechanism for extracellular K<sup>+</sup> clearance described in this study extends the concept of <em>K<sup>+</sup></em> spatial buffering, which was introduced more than a half century ago (Kofuji & Newman, 2004; Orkand <em>et al</em>., 1966). There have been several experimental, modeling, and analytic studies of K<sup>+</sup> spatial buffering since then (Gardner-Medwin, 1983,Gardner-Medwin & Nicholson, 1983; Chen & Nicholson, 2000). However, the classic description does not take into account isopotentiality of the astrocyte syncytium. With sufficiently strong and widespread gap junction coupling, astrocytes near the region of elevated K<sup>+</sup> concentration do not depolarize significantly and this provides a powerful driving force, , for K<sup>+ </sup>uptake through membrane K<sup>+</sup> channels. This important role of syncytial isopotentiality was speculated in previous papers (Muller, 1996); however, this was not experimentally demonstrated until (Ma <em>et al</em>., 2016). As demonstrated in (Ma <em>et al</em>., 2016) syncytial isopotentiality minimizes the local high K<sub>e</sub>-induced <em>V<sub>A</sub></em> depolarization, and this maintains a sustained driving force for K<sup>+</sup> uptake. By extension, syncytial isopotentiality also increases the driving force for K<sup>+</sup> release in distant regions where <em>K<sub>e</sub></em> remains at the physiological level. Additionally, increase in both driving forces creates a maximum driving force for intracellular K<sup>+</sup> transfer from a high K<sup>+</sup> region to remote regions with normal K<sup>+</sup>. Therefore, syncytial isopotentiality facilitates all three critical steps in K<sup>+</sup> spatial buffering: K<sup>+</sup> uptake, intercellular transfer and release (Kofuji & Newman, 2004). Furthermore, recent experiments demonstrate that syncytial isopotentiality arises in several regions throughout the central nervous system and may be a unified mechanism governing the operation of astrocyte networks (Kiyoshi <em>et al.</em>, 2019; Kiyoshi & Zhou, 2019; Huang <em>et al.</em>, 2018).</p>
<p>Some papers have proposed that the inward rectifying K<sup>+</sup> current, <em>I<sub>Kir</sub></em>, is primarily responsible for K<sup>+</sup> buffering. A computational model developed in (Sibille <em>et al</em>., 2015), for example, suggests that astrocytic Kir4.1 channels are sufficient to account for elevated extracellular K<sup>+</sup> clearance, even without gap junctional coupling. However, there are important differences between the model presented in (Sibille <em>et al</em>., 2015) and that developed in this paper. In particular, the astrocyte membrane equation in (Sibille <em>et al</em>., 2015) contains a nonspecific leak current with a fixed reversal potential that helps stabilize the astrocyte membrane potential at -80 mV. This keeps the astrocyte’s membrane potential sufficiently hyperpolarized during neuronal activity so that <em>I<sub>Kir </sub></em>can reverse to an inward current. In our model, the astrocyte’s membrane potential remains hyperpolarized due to gap junctions and isopotentiality.</p>
<p>There have been numerous earlier papers that have addressed various issues related to modeling signals in astrocyte-neuronal networks. In particular, several papers have introduced models for spreading depolarizations, spreading depression, epilepsy, persistent activity and the propagation of Ca<sup>2+</sup> waves (Cressman <em>et al</em>., 2009; Frohlich & Bazhenov, 2006; Hubel & Dahlemm 2014; Hubel <em>et al</em>., 2014; Huguet <em>et al</em>., 2016; Kager <em>et al</em>., 2002; O’Connell & Mori, 2016; Somjen <em>et al</em>., 2008; Ullah <em>et al</em>., 2009; Wei <em>et al</em>., 2014; Zandt <em>et al</em>., 2011). Moreover, several papers have used dynamical systems methods to reduce the complexity of the models and analyze how the neuronal spiking activity depends on the astrocytes’ ability to maintain ion homeostasis (Barreto & Cressman, 2011; Cressman <em>et al</em>., 2009; Frohlich & Bazhenov, 2006; Oyehaug <em>et al</em>., 2012; Zandt <em>et al</em>., 2011). Many of these previous models assumed that the role of the astrocytes is to simply buffer extracellular K<sup>+</sup>; this was modeled by including a simple buffering term in the equation for extracellular K<sup>+</sup>. We have built on and extended previous modeling studies by incorporating a detailed biophysical model for the astrocytes, considering the role played by gap junctions and reducing a model for the response of neurons to excitatory input to a one dimensional map.</p>
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</div></div></div><div class="field field-name-field-acknowledgments field-type-text-long field-label-above"><div class="field-label">Acknowledgments: </div><div class="field-items"><div class="field-item even"><p>This work was partially supported by NIH awards RO1NS062784 and R56NS097972. </p>
<p> </p>
</div></div></div><div class="field field-name-field-references field-type-text-long field-label-above"><div class="field-label">References: </div><div class="field-items"><div class="field-item even"><p class="rtejustify">Allaman I, Belanger M, Magistretti PJ. 2011. Astrocyte-neuron metabolic relationships: for better and for worse. Trends Neurosci 34:76-87.<br />
Amiri M, Hosseinmardi N, Bahrami F, Janahmadi M. 2013. Astrocyte- neuron interaction as a mechanism responsible for generation of neural synchrony: a study based on modeling and experiments. J Comput Neurosci 34:489-504.<br />
Attwell D, Buchan AM, Charpak S, Lauritzen M, Macvicar BA, Newman EA. 2010. Glial and neuronal control of brain blood flow. Nature 468:232-43.<br />
Barres BA. 2008. The mystery and magic of glia: a perspective on their roles in health and disease. Neuron 60:430-40.<br />
Barreto E, Cressman JR. 2011. Ion concentration dynamics as a mechanism for neuronal bursting. J Biol Phys 37:361-73.<br />
Bellot-Saez A, Kekesi O, Morley JW, Buskila Y. 2017. Astrocytic modulation of neuronal excitability through K(+) spatial buffering. Neurosci Biobehav Rev 77:87-97.<br />
Chen KC, Nicholson C. 2000. Spatial buffering of potassium ions in brain extracellular space. Biophys J 78:2776-97.<br />
Chen Y, Swanson RA. 2003. Astrocytes and brain injury. J Cereb Blood Flow Metab 23:137-49.<br />
Chever O, Dossi E, Pannasch U, Derangeon M, Rouach N. 2016. Astroglial networks promote neuronal coordination. Sci Signal 9:ra6.<br />
Cressman JR, Jr., Ullah G, Ziburkus J, Schiff SJ, Barreto E. 2009. The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics. J Comput Neurosci 26:159-70.<br />
Dirnagl U, Iadecola C, Moskowitz MA. 1999. Pathobiology of ischaemic stroke: an integrated view. Trends Neurosci 22:391-7.<br />
Ermentrout B. 2002. Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. Society for Industrial and Applied Mathematics.<br />
Ermentrout GB, Terman DH. 2010. Mathematical Foundations of Neuroscience (Springer Science & Business Media).<br />
Frohlich F, Bazhenov M. 2006. Coexistence of tonic firing and bursting in cortical neurons. Phys Rev E Stat Nonlin Soft Matter Phys 74:031922.<br />
Gardner-Medwin AR. 1983. Analysis of potassium dynamics in mammalian brain tissue. J Physiol 335:393-426.<br />
Gardner-Medwin AR, Nicholson C. 1983. Changes of extracellular potassium activity induced by electric current through brain tissue in the rat. J Physiol 335:375-92.<br />
Giaume C, Koulakoff A, Roux L, Holcman D, Rouach N. 2010. Astroglial networks: a step further in neuroglial and gliovascular interactions. Nat Rev Neurosci 11:87-99.<br />
Haydon PG, Carmignoto G. 2006. Astrocyte control of synaptic transmission and neurovascular coupling. Physiol Rev 86:1009-31.<br />
Huang M, Du Y, Kiyoshi CM, Wu X, Askwith CC, McTigue DM, Zhou M. 2018. Syncytial isopotentiality: an electrical feature of spinal cord astrocyte networks. Neuroglia 1:271-279.<br />
Hubel N, Dahlem MA. 2014. Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs. PLoS Comput Biol 10:e1003941.<br />
Hubel N, Scholl E, Dahlem MA. 2014. Bistable dynamics underlying excitability of ion homeostasis in neuron models. PLoS Comput Biol 10:e1003551.<br />
Huguet G, Joglekar A, Messi LM, Buckalew R, Wong S, Terman D. 2016. Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model. Biophys J 111:452-462.<br />
Iadecola C, Nedergaard M. 2007. Glial regulation of the cerebral microvasculature. Nat Neurosci 10:1369-76.<br />
Kager H, Wadman WJ, Somjen GG. 2002. Conditions for the triggering of spreading depression studied with computer simulations. J Neurophysiol 88:2700-12.<br />
Kimelberg HK, Nedergaard M. 2010. Functions of astrocytes and their potential as therapeutic targets. Neurotherapeutics 7:338-53.<br />
Kiyoshi CM, Du Y, Zhong S, Wang W, Taylor AT, Xiong B, Ma B, Terman D, Zhou M. 2018. Syncytial isopotentiality: a system-wide electrical feature of astrocytic networks in the brain. Glia.<br />
Kiyoshi CM, Zhou M. 2019. Astrocyte syncytium: a functional reticular system in the brain. Neural Regen Res 14:595-596.<br />
Kofuji P, Newman EA. 2004. Potassium buffering in the central nervous system. Neuroscience 129:1045-56.<br />
Ma B, Buckalew R, Du Y, Kiyoshi CM, Alford CC, Wang W, McTigue DM, Enyeart JJ, Terman D, Zhou M. 2016. Gap junction coupling confers isopotentiality on astrocyte syncytium. Glia 64:214-26.<br />
Moskowitz MA, Lo EH, Iadecola C. 2010. The science of stroke: mechanisms in search of treatments. Neuron 67:181-98.<br />
Muller CM. 1996. Gap-junctional communication in mammalian cortical astrocytes: development, modifiability and possible functions. In: Spray DC, Austin DR Gap junctions in the nervous system Editors TX: RG Landes Company pp 203-212.<br />
Nakase T, Fushiki S, Sohl G, Theis M, Willecke K, Naus CC. 2003. Neuroprotective role of astrocytic gap junctions in ischemic stroke. Cell Commun Adhes 10:413-7.<br />
Nedergaard M, Dirnagl U. 2005. Role of glial cells in cerebral ischemia. Glia 50:281-6.<br />
Nedergaard M, Ransom B, Goldman SA. 2003. New roles for astrocytes: redefining the functional architecture of the brain. Trends Neurosci 26:523-30.<br />
Newman EA. 2003. New roles for astrocytes: regulation of synaptic transmission. Trends Neurosci 26:536-42.<br />
O'Connell R, Mori Y. 2016. Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression. Bull Math Biol 78:1943-1967.<br />
Orkand RK, Nicholls JG, Kuffler SW. 1966. Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia. J Neurophysiol 29:788-806.<br />
Oyehaug L, Ostby I, Lloyd CM, Omholt SW, Einevoll GT. 2012. Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms. J Comput Neurosci 32:147-65.<br />
Pannasch U, Vargova L, Reingruber J, Ezan P, Holcman D, Giaume C, Sykova E, Rouach N. 2011. Astroglial networks scale synaptic activity and plasticity. Proc Natl Acad Sci U S A 108:8467-72.<br />
Poskanzer KE, Yuste R. 2016. Astrocytes regulate cortical state switching in vivo. Proc Natl Acad Sci U S A 113:E2675-84.<br />
Ransom BR. 1996. Do glial gap junctions play a role in extracellular ion homeostasis? In: Spray DC, Austin DR Gap junctions in the nervous system Editors TX: RG Landes Company pp 159-173.<br />
Ransom CB, Sontheimer H. 1995. Biophysical and pharmacological characterization of inwardly rectifying K+ currents in rat spinal cord astrocytes. J Neurophysiol 73:333-46.<br />
Sibille J, Dao Duc K, Holcman D, Rouach N. 2015. The neuroglial potassium cycle during neurotransmission: role of Kir4.1 channels. PLoS Comput Biol 11:e1004137.<br />
Somjen GG. 2001. Mechanisms of spreading depression and hypoxic spreading depression-like depolarization. Physiol Rev 81:1065-96.<br />
Somjen GG, Kager H, Wadman WJ. 2008. Computer simulations of neuron-glia interactions mediated by ion flux. J Comput Neurosci 25:349-65.<br />
Szabo Z, Heja L, Szalay G, Kekesi O, Furedi A, Szebenyi K, Dobolyi A, Orban TI, Kolacsek O, Tompa T and others. 2017. Extensive astrocyte synchronization advances neuronal coupling in slow wave activity in vivo. Sci Rep 7:6018.<br />
Ullah G, Cressman JR, Jr., Barreto E, Schiff SJ. 2009. The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states. II. Network and glial dynamics. J Comput Neurosci 26:171-83.<br />
Wei Y, Ullah G, Schiff SJ. 2014. Unification of neuronal spikes, seizures, and spreading depression. J Neurosci 34:11733-43.<br />
Zandt BJ, ten Haken B, van Dijk JG, van Putten MJ. 2011. Neural dynamics during anoxia and the "wave of death". PLoS One 6:e22127.<br />
Zhao Y, Rempe DA. 2010. Targeting astrocytes for stroke therapy. Neurotherapeutics 7:439-51.</p>
<p> </p>
</div></div></div>Fri, 29 Mar 2019 20:05:09 +0000Root Addmin250 at http://operamedphys.orghttp://operamedphys.org/content/modeling-role-astrocyte-syncytium-and-k-buffering-maintaining-neuronal-firing-patterns#commentsTemperature Effects on Non-TRP Ion Channels and Neuronal Excitability
http://operamedphys.org/OMP_2017_003_0049
<div class="field field-name-field-authors field-type-taxonomy-term-reference field-label-hidden clearfix"><ul class="links"><li class="taxonomy-term-reference-0"><a href="/authors/sergey-m-korogod">Sergey M. Korogod</a> / <a href="/authors/liliia-e-demianenko">Liliia E. Demianenko</a></li></ul></div><div class="field field-name-field-author-affiliations field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="jqueryui-formatters accordion" data-instance="211-7-9"><h3><a href="#jqueryui-formatters-accordion">Author Affiliations</a></h3><div><p><p>Sergey M. Korogod<sup>1</sup> * and Liliia E. Demianenko<sup>2</sup></p>
<p><sup>1</sup> A. A. Bogomoletz institute of physiology, NAS of Ukraine, 01024 Kiev, Ukraine;<br /><sup>2</sup> Dnipropetrovsk medical academy of Ministry of public health of Ukraine, Dnipro, Ukraine.</p>
</p></div></div></div></div></div><div class="field field-name-field-corresponding-author field-type-text-long field-label-inline clearfix"><div class="field-label">Corresponding author: </div><div class="field-items"><div class="field-item even"><p>Sergey M. Korogod (<a href="mailto:isabroad@gmail.com">isabroad@gmail.com</a>)</p>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><h3 class="field-label">Keywords: </h3><ul class="links inline"><li class="taxonomy-term-reference-0" rel="dc:subject"><a href="/keywords/ion-channels" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ion channels</a></li><li class="taxonomy-term-reference-1" rel="dc:subject"><a href="/keywords/neuron" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">neuron</a></li><li class="taxonomy-term-reference-2" rel="dc:subject"><a href="/keywords/temperature-sensitivity" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">temperature sensitivity</a></li><li class="taxonomy-term-reference-3" rel="dc:subject"><a href="/keywords/excitability" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">excitability</a></li></ul></div><div class="field field-name-field-abstract field-type-text-long field-label-above"><div class="field-label">Abstract: </div><div class="field-items"><div class="field-item even"><p class="rtejustify">This review focuses on temperature dependence of biophysical properties of ion channels of other than TRP types. Some functional consequences of the channels temperature sensitivity for neuronal excitation in health and disease are considered.</p>
</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><div class="rtejustify"><strong><span style="font-size:14px">Introduction</span></strong></div>
<div class="rtejustify"> </div>
<div class="rtejustify">The ability of the mammalian nervous system to detect and discriminate external and internal temperatures is crucial for the homeostasis and survival. Temperature related changes in neurons excitability provide this ability. With discovery of TRP channels specifically sensitive to temperature (thermoTRPs), changes in neuronal excitability became attributed almost exclusively to these channels as they are activated in physiologically important range (0–60 °C) and conduct depolarizing currents. In recent years, the functional role of temperature dependence of other type ion channels is revisited (Vriens, Nilius, & Voets, 2014). Whereas the expression, biophysical properties, and functional roles of the thermoTRPs are matters of numerous reviews (Michael J Caterina & Pang, 2016; Talavera, Nilius, & Voets, 2008; Voets, 2012), the temperature-related properties of the non-TRP channels are not systematically reviewed. Here we (1) consider which and how biophysical properties of non-TRP channels depend on temperature and (2) elucidate how this temperature-dependence influences the neuronal excitability and, consequently, some physiological functions. The scope of this review is limited to the data obtained in warm-blooded animals. We address the following specific questions. Which non-TRP channels significantly change their biophysical properties with temperature? Which specific properties of the channels conducting de- and hyperpolarizing currents are influenced by temperature? In which ways these influences are performed? What are possible functional consequences of normal or altered thermosensitivity of the non-TRP channels in health and disease? </div>
<div class="rtejustify"> </div>
<div class="rtejustify"><span style="font-size:14px"><strong>Voltage-gated channels sensitive to temperature</strong></span></div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Channels conducting depolarization currents</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em>Ca<sup>2+</sup> channels</em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Temperature sensitivity of the biophysical characteristics is described for Cav3.1-3.3 also known as low-voltage activated (LVA), T-type and Cav 1.2 and Cav 1.3, high-voltage activated (HVA), L-type (Chemin et al., 2007; Coulter, Huguenard, & Prince, 1989; Iftinca et al., 2006; Johnson & Marcotti, 2008; Narahashi, Tsunoo, & Yoshii, 1987; Nobile, Carbone, Lux, & Zucker, 1990; Radzicki et al., 2013; Rosen, 1996).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em>Na<sup>+</sup> channels</em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Temperature dependence was found in TTX-sensitive (TTXs, subtypes Nav 1.1 – 1.4; Nav 1.6; Nav 1.7) and TTX-resistant (TTXr, subtypes Nav 1.5; Nav 1.8; and Nav 1.9) sodium channels (Rosen, 2001; Sarria, Ling, & Gu, 2012; Zimmermann et al., 2007). The Nav1.1, Nav1.6, and Nav1.7 channels are also called transient sodium (NaT) channels because of their fast activating and inactivating kinetics. Correspondingly, the non-inactivating or slowly inactivating channels of Nav1.8 and Nav1.9 subtypes are termed persistent sodium (NaP) channels. Occurrence of pathological sensitivity to temperature is described for Nav1.2 with mutated β1-subuinit (Egri, Vilin, & Ruben, 2012).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em>HCN channels</em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Strong temperature dependence was demonstrated for hyperpolarization-activated channels conducting cationic depolarizing currents of HNC1-type influencing the discharge patterns of mammalian cold thermoreceptor endings. Pathological thermosensitivity occurring in mutated HCN2 was shown to contribute to febrile seizures (Nakamura et al., 2013).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Channels conducting hyperpolarization currents</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em>K<sup>+</sup> channels</em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Temperature sensitivity was described for channels conducting A-type potassium currents (Sarria et al., 2012), for the Kv2.1 and Kv4.3 channels with uncoupled voltage sensor (Yang & Zheng, 2014), and for the TREK-1, TREK-2, and TRAAK two-pore channels (Schneider, Anderson, Gracheva, & Bagriantsev, 2014).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em>Cl<sup>−</sup> channels</em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Highly temperature dependent properties are inherent in the anoctamin-1 (ANO1, also known as TMEM16) Ca<em><sup>2+</sup></em>-activated chloride conducting channels (Cho et al., 2012) and CLC proteins mediating Cl− transfer (Pusch & Zifarelli, 2014).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><span style="font-size:14px"><strong>Biophysical properties altered by temperature</strong></span></div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Peak and steady currents, current-voltage relations</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Whole cell voltage-clamp recordings at 21 °C and 37 °C from tsA-201 cells exogenously expressing the three different rat brain T-type calcium channel isoforms, Cav3.1, Cav3.2, and Cav3.3 and the human isoform, Cav3.3h showed that the heating led to increase in Ca<em><sup>2+</sup></em> current amplitude by 1.8-fold for Cav3.1, 2.1-fold for Cav3.2, 1.6-fold for Cav3.3, and 1.8-fold for Cav3.3h (Iftinca et al., 2006). The effects were the same when temperature was either ramped up from 21 °C, or down from 37 °C. Calcium current of L-type (predominantly Cav1.3) recorded from the inner hear cells (IHCs) of immature (<P12) and adult (>P20) gerbils at body temperature (35–37 °C) was significantly greater in magnitude than that recorded at room temperature (21–23 °C) with the temperature ratios Q10 =1.9 and 1.4 for immature and adult animals, respectively. Similar temperature dependence was observed for the steady current through L-type (Cav1.3) channels in IHCs of 2-week-old C57Bl6/N mice (Nouvian, 2007) with the temperature ratios of Q10 =1.1 (whole-cell recordings) or Q10 =1.3 (perforated patch). The significantly steeper temperature dependence of this current in immature IHCs suggests existence of developmental change in these properties. Based on the singe-channel recordings from dissociated dorsal root ganglion (DRG) neurons of 12-day old chicken embryo, it is suggested that the larger Ca<em><sup>2+</sup></em> current at physiological temperature is likely due to an increased opening probability (Acerbo & Nobile, 1994).</div>
<div class="rtejustify">Cooling from 30 °C to 10 °C reduced peak values of both TTX-sensitive and TTX-resistant sodium currents (through channels of Nav1.7 and Nav1.8 subtypes, respectively), recorded in cultured small-size DRG neurons of rat and heterologous cells expressing these channels (Sarria et al., 2012; Zimmermann et al., 2007). The decrease in current amplitudes on cooling was significantly stronger for Nav1.7 than for Nav1.8, especially at a membrane potential of -80 mV, which is similar to the physiological membrane potential. Study of sodium current through TTXs channels of GH3 cells from rat pituitary showed absence of noticeable change in the current-voltage relationship with temperature changing in the range from 23 to 37 °C (Rosen, 2001). In Chinese Hamster Ovary (CHO) cells expressing Nav1.2, particularly with beta1 subunit, it was found that elevation of temperature from 22 to 34 °C causes increase in the persistent and window current (Egri et al., 2012). </div>
<div class="rtejustify">A-type K+ currents (IA) were strongly inhibited by cooling, whereas non-inactivating K+ currents (IK) were weakly inhibited by this cue (Sarria et al., 2012). Experiments on HEK293 cells expressing Kv2.1 and Kv4.3 channels with the use of the voltage ramp protocol (−100 to +50 mV in 300 ms) revealed a large increase in peak currents through those channels with heating from 21 °C to 33 °C (Kv2.1) or 23 °C to 39 °C (Kv4.3) (Yang & Zheng, 2014). This was observed within a relatively narrow voltage range (near – 50 mV). In this range the heating-induced increase in Kv2.1 and Kv4.3 conductance was characterized by a transient increase in Q10 to very high values of about 20 to 30 that is more characteristic of specialized thermosensitive channels of TRP family. At normal body temperature, the two-pore potassium (K2P) channels TREK-1 (K2P2.1/KCNK2), TREK-2 (K2P10.1/KCNK10), and TRAAK (K2P4.1/KCNK2) regulate cellular excitability by providing voltage-independent leak of potassium. Heat dramatically potentiates K2P channel activity and further affects excitation (Schneider et al., 2014). At room temperature, TREK-1 exhibits only background potassium leak, which increases with temperature, reaching maximum at ~42 °C. Interestingly, TREK-1 has its half-maximal temperature activation point (T1/2) at ~37 °C, implying that the midpoint of the channel's dynamic range is centered on the homeostatic thermal set point for most mammals. </div>
<div class="rtejustify">Chloride current through ANO1 increased by heating with a steeper temperature dependence at higher intracellular Ca<em><sup>2+</sup></em> concentrations (Cho et al., 2012). Notably, raising the temperature above 44 °C resulted in robust inward Cl− currents at a holding potential of −60 mV in ANO1-HEK cells. Heat-evoked ANO1 currents in these cells were outwardly rectifying and reversed at near zero mV (−5.0 mV, n = 5).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Voltage-dependence of activation and inactivation</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">In the above mentioned experiments on Cav3.3 isoforms (Iftinca et al., 2006), heating from 21 °C and 37 °C caused a small (~9 mV) but statistically significant hyperpolarizing shift of the voltage-dependent activation for the Cav3.1 (the half-activation potentials Va1/2 changed from –52.4±0.7 to –60.5±0.9 mV) and Cav3.2 (from –42.9±0.8 to –51.5±1.0 mV); however, the shift was insignificant for rat Cav3.3 (from –72.9±1.1 to –73.5±1.3 mV) or human Cav3.3h (from –55.5±1.1 to –56.8±0.9 mV). The temperature ratio (Q) of the Va1/2 values calculated at the two temperatures for Cav3.1 and Cav3.2 was Q=1.15 and 1.2, respectively. Neither isoform displayed significant change in the slope of its activation voltage-dependence. In the same experiments, heating also shifted the voltage-dependent inactivation. There was a hyperpolarization shift in half-inactivation potential Vh1/2 for Cav3.1 (from –72.2±1.1 to –75.4±4.3 mV), Cav3.2 (from –64.2±2.2 to –73.7±1.2 mV), and Cav3.3h (from –69.0±1.9 to –82.1±1.5 mV) and a small depolarizing shift for the rat Cav3.3 (from –78.3±0.8 to –73.4±2.5 mV). The effects were statistically significant only for the Cav3.2 subtype and Cav3.3h (with temperature ratios Q=1.15 and 1.19, respectively). The combined effects of temperature on half activation and half inactivation potentials further resulted in a hyperpolarizing shift in the position of the window current (i.e. the overlap between activation and inactivation curves) for Cav3.1 and Cav3.2, which would allow these channels to be active at neuronal resting membrane potentials (Perez-Reyes, 2003).</div>
<div class="rtejustify">Sodium currents, both TTX-sensitive (TTXs) and TTX-resistant, also displayed a small shift of the voltage-dependent activation towards more hyperpolarized potentials, however this was caused by cooling from ~30 to 10°C. This was observed in the mentioned above experiments on cultured DRG neurons expressing Nav1.7 (TTXs) and Nav1.8 (TTXr) channels (Zimmermann et al., 2007). In the same experiments, cooling had little effect on fast inactivation but markedly shifted the slow inactivation of TTXs currents towards more hyperpolarized potentials. In contrast, slow inactivation of TTXr currents was resistant to cooling. These distinct properties of TTXs and TTXr currents were conserved in recombinant Nav1.7 (TTXs) and Nav1.8 (TTXr) channels; that is, cooling resulted in a leftward shift of slow inactivation of Nav1.7 but not of Nav1.8 (Zimmermann et al., 2007). A consistent result was obtained in other studies on rat DRG neurons (Sarria et al., 2012), in which steady-state slow inactivation curve of TTXs channel shifted to more hyperpolarizing voltages with cooling that resulted in substantial slow inactivation near resting membrane potentials. In CHO cells expressing Nav1.2 β1 subunit, it was found that elevation of temperature from 22 to 34°C causes a hyperpolarization shift of the voltage-dependent activation for all subunits (Egri et al., 2012). The same effect was found for the fast inactivation. The rate of fast inactivation recovery was also increased by elevated temperature for all subunits. </div>
<div class="rtejustify">The above mentioned heat-induced augmentation of potassium currents through Kv2.1 and Kv4.3 channels was not due to a shift of the half-activation potential V1/2 as the peak Q10 values exhibited no correlation with the temperature-dependent shifts in V1/2 (Yang & Zheng, 2014).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><strong><em>Rates (time constants) of voltage-dependent activation and inactivation</em></strong></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Increasing the temperature from 21 °C to 37 °C led to a large and statistically significant reduction in the time constants of both activation and inactivation for rat Cav3.1, Cav3.2, Cav3.3 and human Cav3.3h isoforms of Cav3 channels explored in the above-mentioned study (Iftinca et al., 2006). The accelerating effect of such heating on the inactivation kinetics of Cav3.1 and Cav 3.3 was apparent in a wide range of voltages, but in case of Cav 3.2 it was observed only below –20 mV. Noteworthy, the inactivation time course was well described by a single exponential at 21 °C whereas at 37 °C it became bi-exponential because of occurrence of a second slower decay component. Absence of the latter component at 21 °C is particularly notable as it has often been considered as a feature of this T-type channel isoform. Nonetheless, Cav3.3 channels still inactivated more slowly than the other T-type channel subtypes (Iftinca et al., 2006). In view of its significance for duration of the refractory period after neuron APs termination, the temperature dependence of the time required for recovery from inactivation is of particular interest. The time constant of this process significantly decreased with heating from 21 °C to 37 °C as follows from measurements performed at –120 mV for Cav3.1 (Q=0.4), Cav3.2 (Q=0.35), rat Cav3.3 (Q=0.33), and human Cav3.3h (Q=0.3). At 37 °C, the recovery times were similar for all three isoforms suggesting their similar ability to recover from the inactivated state at physiological temperatures. Recovery from inactivation examined at other membrane potentials (–80 and –100 mV for Cav3.1 and Cav3.2; and –90 and –110 mV for Cav3.3) was also significantly accelerated at 37 °C irrespective of membrane potential. However, unlike at more negative recovery potentials, at –80 mV Cav3.2 recovered from inactivation more slowly than Cav3.1. At 21 °C and 37 °C, the recovery time constants for Cav3.1 were 444.1 ±76.2 ms and 197.3 ±15.5 ms, respectively (Q=0.44) and for Cav3.2 they were 722.9 ±73.0 ms and 488.7 ±48.7 ms, respectively (Q=0.68). The recovery could not be examined for rat Cav3.3 at –80 mV because of its more negative activation and inactivation range. For the same three isoforms of Cav3 channels, time constants of deactivation were determined from the tail currents recorded at 21 °C at 37 °C and fitted with single exponential functions. At both temperatures, they were similar for Cav3.2 regardless of the membrane potentials (–120, –100, and –80 mV) and became significantly different at more depolarized potentials for Cav3.1 and rat Cav3.3 (Fig. 4B, D). Interestingly, at both temperatures the rat and human Cav3.3 channel showed the slowest deactivation kinetics compared with the other two isoforms irrespective of the holding potential, suggesting that this channel subtype may have an energetically more stable open conformation. At 37 °C, the deactivation time constants were comparable for the three rat isoforms at potentials more negative than –100 mV, but rat Cav3.3 was substantially slower to deactivate at typical neuronal resting potentials. Moreover, the values obtained for Cav3.3h channels were smaller than those seen with the rat Cav3.3 isoforms, but still showed temperature dependence (Q=1.15 for Cav3.1; Q=0.83 for Cav3.2; Q=0.56 for Cav3.3; and Q=0.42 for Cav3.3h). Studies of calcium L-type (Cav1.3) current in mouse IHCs showed that with lowering the temperature from~37 °C to room temperature the activation time constant was slowed down by about half (Marcotti, Johnson, Rusch, & Kros, 2003; Nouvian, 2007). Measured at -20 mV average time constant of activation τm decreased (activation accelerated) with temperature increase in the range from 23 °C to 37 °C with sharper slope (greater temperature sensitivity) in the lower temperature sub-range 23-28 °C compared to higher temperature sub-range. This feature was especially evident at lower membrane voltages. The inactivation time constant τh of this current linearly decreased (inactivation accelerated too) in the whole mentioned temperature range (Egri et al., 2012; Rosen, 2001).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><a href="http://operamedphys.org/sites/default/files/pictures/articles/2017_issue3-4_volume3/Korogod/fig_1.jpg" target="_blank"><img alt="Figure 1. Temperature influences the Nernst equilibrium potential (A), steady-state kinetics (B) and time constants (C) of activation and inactivation, recovery from inactivation (D), and, possibly, other yet unknown biophysical properties of channels." id="Figure 1. Temperature influences the Nernst equilibrium potential (A), steady-state kinetics (B) and time constants (C) of activation and inactivation, recovery from inactivation (D), and, possibly, other yet unknown biophysical properties of channels." longdesc="Figure 1. Temperature influences the Nernst equilibrium potential (A), steady-state kinetics (B) and time constants (C) of activation and inactivation, recovery from inactivation (D), and, possibly, other yet unknown biophysical properties of channels." src="http://operamedphys.org/sites/default/files/pictures/articles/2017_issue3-4_volume3/Korogod/fig_1.jpg" style="height:314px; margin-left:150px; margin-right:150px; width:600px" title="Figure 1. Temperature influences the Nernst equilibrium potential (A), steady-state kinetics (B) and time constants (C) of activation and inactivation, recovery from inactivation (D), and, possibly, other yet unknown biophysical properties of channels." /></a></div>
<div class="rtejustify"><span style="font-size:11px"><strong>Figure 1.</strong> Temperature influences the Nernst equilibrium potential (A), steady-state kinetics (B) and time constants (C) of activation and inactivation, recovery from inactivation (D), and, possibly, other yet unknown biophysical properties of channels.</span></div>
<div class="rtejustify"> </div>
<div class="rtejustify"> </div>
<div class="rtejustify"><span style="font-size:14px"><strong>Functional implications of channels thermosensitive properties in health and disease</strong></span></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Temperature dependence is inherent in practically all type ion currents as corresponding current-conducting channels change their biophysical properties with temperature. These changes are quantitatively characterized by the temperature coefficient Q10. It can be calculated as the relative increase in current amplitude at a specific voltage when temperature increases by 10 degrees, i.e. as Q10 = IT+10/IT. Q10≥5 was proposed as a criterion for distinguishing thermosensitive TRP channels (thermoTRPs) from the whole TRP family (Voets, 2012). According this criterion some non-TRP channels also can be have qualified as highly thermosensitive as they have Q10≥5 (e.g. CLC , ANO1 (Pusch & Zifarelli, 2014). Unlike channels, there is no generally admitted criterion for distinguishing a thermosensory unit at the level of individual neurons. We consider that expression of channels with so high Q10 by itself is necessary but not sufficient for a neuron to function as a thermosensory unit. Our suggestion is supported by known expression of highly thermosensitive TRPV1 and TRPM8 channels in neurons, which do not function as thermosensors. Expression of the high-Q10 channels must be sufficiently high for providing a steep temperature dependence of a neuron reaction (e.g. firing rate) to changes in temperature. Neurons, which are not high-thermosensitive in health can become such in result of channelopathy that leads to development and maintenance of pathological states (febrile seizures). However, implications of existing temperature dependence of CLC proteins, which mediate Cl− transport, are unknown (Pusch & Zifarelli, 2014). This is also true in case of mechanosensitive Piezo-1 and Piezo-2 channels clearly displaying temperature dependence (Coste et al., 2010).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><span style="font-size:14px"><strong>Physiological functions</strong></span></div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Central thermodetection</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Participation of TRP channels in thermoregulatory responses of neurons in the preoptic area and anterior hypothalamus (PO/AH) is disputed. On the one hand, mechanism for PO/AH neuronal warm sensitivity was attributed to thermosensitive TRPV4 channels (Caterina et al., 1997; Güler et al., 2002; Patapoutian, Peier, Story, & Viswanath, 2003). On the other hand, Wechselberger and co-authors (Wechselberger, Wright, Bishop, & Boulant, 2006) showed the absence of TRPV4 channels expression in PO/AH neurons and, in the simulation study, they demonstrated that the potassium leak and hyperpolarization-activated cationic channels (HCN) determined warm-induced responses of PO/AH neurons. Recent study (Kamm & Siemens, 2017) demonstrated that other type TRP-channel, namely TRPM2 is involved in thermoregulatory responses of PO/AH neurons. They showed that heat (>37 °C) responses of POA neurons were significantly reduced in TRPM2-deficient mice compared to wild-type ones. Two-pore potassium channels, namely TREK are also involved in thermoregulatory responses of neurons in PO/AH. TREK are activated by heating and thus inhibit depolarization of these PO/AH neurons (Maingret et al., 2000).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><em><strong>Peripheral thermodetection</strong></em></div>
<div class="rtejustify"> </div>
<div class="rtejustify">Cold reception. It is generally thought that activity of TRPM8 and TRPA1 channels underlies sensitivity of PANs to innocuous and noxious cold, respectively, while the role of other channels is underestimated. For instance, in cold nociceptor neurons, cooling progressively enhances the voltage-dependent slow inactivation of TTX-sensitive sodium channels, whereas the inactivation properties of TTX-insensitive Nav1.8 channels are entirely cold-resistant. The observation that Nav1.8-deficient mice show negligible responses to noxious cold and mechanical stimulation at low temperatures suggests that Nav1.8 remains available as the sole electrical impulse generator in nociceptors that transmits nociceptive information to CNS (Zimmermann et al., 2007). Decreased activation of TREK-1 potassium channels at low temperatures reduces their contribution of the membrane hyperpolarization and thus promotes the depolarization of receptor endings of cold PANs (Maingret et al., 2000).</div>
<div class="rtejustify">Warm reception. Activity of vast majority of ion channels is increased and/or accelerated with increasing temperature. This was shown for practically all type channels conducting the depolarizing currents. In case of the hyperpolarizing current conducting channels this property was demonstrated only for the two-pore potassium (K2P) ones. Moreover, K2Ps may lose selectivity to potassium under physiologically relevant conditions that may turn them into a heat-activated excitatory ion channel similar to TRPV1, i.e. its own functional antipode. In PANs from TRV1-KO mice, APs generation in response to ramp-heating was attributed to presence of high temperature-sensitive (Q10 ~20) Ca<em><sup>2+</sup></em> activated chloride channels of ANO1-type (Cho et al., 2012).</div>
<div class="rtejustify"> </div>
<div class="rtejustify"><span style="font-size:14px"><strong>Role in neuropathology</strong></span></div>
<div class="rtejustify"> </div>
<div class="rtejustify">The role of ion channels of other than thermoTRP types in temperature dependence of pathological processes was shown for development and treatment of neuronal hyperexcitability. On the one hand, NaP with mutated β1 (C121W) subunit shows high temperature sensitivity by widening window current which is one of the possible mechanism of febrile seizures (Egri et al., 2012). On the other hand, it’s known that therapeutic hypothermia suppresses excessive excitability of cortical neurons involved in generation of epileptiform EEG patterns (Motamedi, Gonzalez-Sulser, Dzakpasu, & Vicini, 2012). In our recent model study (Demianenko, Poddubnaya, Makedonsky, Kulagina, & Korogod, 2017) we demonstrated cooling induced deactivation of termoTRP can be a mechanism of suppression of bursting activity of neurons underlying synchronized epileptiform network activity. But role of other non- thermoTRPs is not sufficiently disclosed until now. Heat, voltage, and Ca<em><sup>2+</sup></em> synergistically sensitize each other’s effects on ANO1. Normally, ANO1 are activated by noxious heat. However, temperatures lower than 44 °C also can activate ANO1 at physiological membrane potentials because [Ca<em><sup>2+</sup></em>]i increases under pathological conditions such as inflammation (Cho et al., 2012).</div>
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</div></div></div><div class="field field-name-field-references field-type-text-long field-label-above"><div class="field-label">References: </div><div class="field-items"><div class="field-item even"><p class="rtejustify"> </p>
<p class="rtejustify">Acerbo, P., & Nobile, M. (1994). Temperature dependence of multiple high voltage activated Ca2+ channels in chick sensory neurones. European Biophysics Journal : EBJ, 23, 189–95.<br />
Caterina, M. J., & Pang, Z. (2016). TRP Channels in Skin Biology and Pathophysiology. Pharmaceuticals (Basel, Switzerland), 9. doi:10.3390/ph9040077<br />
Caterina, M. J., Schumacher, M. A., Tominaga, M., Rosen, T. A., Levine, J. D., & Julius, D. (1997). The capsaicin receptor: a heat-activated ion channel in the pain pathway. Nature, 389, 816–24.<br />
Chemin, J., Mezghrani, A., Bidaud, I., Dupasquier, S., Marger, F., Barrère, C., … Lory, P. (2007). Temperature-dependent modulation of CaV3 T-type calcium channels by protein kinases C and A in mammalian cells. The Journal of Biological Chemistry, 282, 32710–8.<br />
Cho, H., Yang, Y. D., Lee, J., Lee, B., Kim, T., Jang, Y., … Oh, U. (2012). The calcium-activated chloride channel anoctamin 1 acts as a heat sensor in nociceptive neurons. Nature Neuroscience, 15, 1015–21.<br />
Coste, B., Mathur, J., Schmidt, M., Earley, T. J., Ranade, S., Petrus, M. J., … Patapoutian, A. (2010). Piezo1 and Piezo2 are essential components of distinct mechanically activated cation channels. Science (New York, N.Y.), 330, 55–60.<br />
Coulter, D. A., Huguenard, J. R., & Prince, D. A. (1989). Calcium currents in rat thalamocortical relay neurones: kinetic properties of the transient, low-threshold current. The Journal of Physiology, 414, 587–604.<br />
Demianenko, L. E., Poddubnaya, E. P., Makedonsky, I. A., Kulagina, I. B., & Korogod, S. M. (2017). Hypothermic Suppression of Epileptiform Bursting Activity of a Hyppocampal Granule Neuron Possessing Thermosensitive TRP Channels (a Model Study: Biophysical and Clinical Aspects). Neurophysiology, 49, 8–18.<br />
Destexhe, A. (2014) Thermodynamic Models of Ion Channels. In: Encyclopedia of Computational Neuroscience., Jaeger, D. and Jung, R. (Eds.), Springer-Verlag: New York, DOI: 10.1007/978-1-4614-7320-6_136-1 <br />
Egri, C., Vilin, Y. Y., & Ruben, P. C. (2012). A thermoprotective role of the sodium channel β1 subunit is lost with the β1 (C121W) mutation. Epilepsia, 53, 494–505.<br />
Eyring, H. (1935) The activated complex in chemical reactions. The Journal of Chemical Physics, 3, 107–115.<br />
Güler, A. D., Lee, H., Iida, T., Shimizu, I., Tominaga, M., & Caterina, M. (2002). Heat-evoked activation of the ion channel, TRPV4. Journal of Neuroscience, 22, 6408–14.<br />
Iftinca, M., McKay, B. E., Snutch, T. P., McRory, J. E., Turner, R. W., & Zamponi, G. W. (2006). Temperature dependence of T-type calcium channel gating. Neuroscience, 142, 1031–42. <br />
Johnson, F. H., Eyring, H., Stover, B. J. (1974) The theory of rate processes in biology and medicine. Wiley: New York.<br />
Johnson, S. L., & Marcotti, W. (2008). Biophysical properties of CaV1.3 calcium channels in gerbil inner hair cells. The Journal of Physiology, 586, 1029–42.<br />
Kamm, G. B., & Siemens, J. (2017). The TRPM2 channel in temperature detection and thermoregulation. Temperature (Austin, Tex.), 4, 21–23.<br />
Maingret, F., Lauritzen, I., Patel, A. J., Heurteaux, C., Reyes, R., Lesage, F., … Honoré, E. (2000). TREK-1 is a heat-activated background K(+) channel. The EMBO Journal, 19, 2483–91.<br />
Marcotti, W., Johnson, S. L., Rusch, A., & Kros, C. J. (2003). Sodium and calcium currents shape action potentials in immature mouse inner hair cells. The Journal of Physiology, 552, 743–61.<br />
Motamedi, G. K., Gonzalez-Sulser, A., Dzakpasu, R., & Vicini, S. (2012). Cellular mechanisms of desynchronizing effects of hypothermia in an in vitro epilepsy model. Neurotherapeutics : The Journal of the American Society for Experimental NeuroTherapeutics, 9, 199–209.<br />
Nakamura, Y., Shi, X., Numata, T., Mori, Y., Inoue, R., Lossin, C., … Hirose, S. (2013). Novel HCN2 mutation contributes to febrile seizures by shifting the channel’s kinetics in a temperature-dependent manner. PloS One, 8, e80376.<br />
Narahashi, T., Tsunoo, A., & Yoshii, M. (1987). Characterization of two types of calcium channels in mouse neuroblastoma cells. The Journal of Physiology, 383, 231–49.<br />
Nobile, M., Carbone, E., Lux, H. D., & Zucker, H. (1990). Temperature sensitivity of Ca currents in chick sensory neurones. Pflugers Archiv : European Journal of Physiology, 415, 658–63.<br />
Nouvian, R. (2007). Temperature enhances exocytosis efficiency at the mouse inner hair cell ribbon synapse. The Journal of Physiology, 584, 535–42.<br />
Patapoutian, A., Peier, A. M., Story, G. M., & Viswanath, V. (2003). ThermoTRP channels and beyond: mechanisms of temperature sensation. Nature Reviews. Neuroscience, 4, 529–39.<br />
Perez-Reyes, E. (2003). Molecular physiology of low-voltage-activated T-type calcium channels. Physiological Reviews, 83, 117–61.<br />
Pusch, M., & Zifarelli, G. (2014). Thermal sensitivity of CLC and TMEM16 chloride channels and transporters. Current Topics in Membranes, 74, 213–31.<br />
Radzicki, D., Yau, H.-J., Pollema-Mays, S. L., Mlsna, L., Cho, K., Koh, S., & Martina, M. (2013). Temperature-sensitive Cav1.2 calcium channels support intrinsic firing of pyramidal neurons and provide a target for the treatment of febrile seizures. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 33, 9920–31.<br />
Rosen, A. D. (1996). Temperature modulation of calcium channel function in GH3 cells. The American Journal of Physiology, 271, C863-8.<br />
Rosen, A. D. (2001). Nonlinear temperature modulation of sodium channel kinetics in GH(3) cells. Biochimica et Biophysica Acta, 1511, 391–6.<br />
Sarria, I., Ling, J., & Gu, J. G. (2012). Thermal sensitivity of voltage-gated Na+ channels and A-type K+ channels contributes to somatosensory neuron excitability at cooling temperatures. Journal of Neurochemistry, 122, 1145–54.<br />
Schneider, E. R., Anderson, E. O., Gracheva, E. O., & Bagriantsev, S. N. (2014). Temperature Sensitivity of Two-Pore (K2P) Potassium Channels. Current Topics in Membranes, 74. doi:10.1016/B978-0-12-800181-3.00005-1<br />
Talavera, K., Nilius, B., & Voets, T. (2008). Neuronal TRP channels: thermometers, pathfinders and life-savers. Trends in Neurosciences, 31, 287–95.<br />
Voets, T. (2012). Quantifying and Modeling the Temperature-Dependent Gating of TRP Channels. In Reviews of Physiology, Biochemistry and Pharmacology 162 (pp. 91–119). Berlin, Heidelberg: Springer Berlin Heidelberg.<br />
Vriens, J., Nilius, B., & Voets, T. (2014). Peripheral thermosensation in mammals. Nature Reviews. Neuroscience, 15, 573–89.<br />
Wechselberger, M., Wright, C. L., Bishop, G. A., & Boulant, J. A. (2006). Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity. American Journal of Physiology. Regulatory, Integrative and Comparative Physiology, 291, R518-29.<br />
Yang, F., & Zheng, J. (2014). High temperature sensitivity is intrinsic to voltage-gated potassium channels. eLife. doi:10.7554/eLife.03255<br />
Zimmermann, K., Leffler, A., Babes, A., Cendan, C. M., Carr, R. W., Kobayashi, J., … Reeh, P. W. (2007). Sensory neuron sodium channel Nav1.8 is essential for pain at low temperatures. Nature, 447, 855–8.</p>
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