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Mechanisms of firing patterns in fast-spiking cortical interneurons.

Golomb D, Donner K, Shacham L, Shlosberg D, Amitai Y, Hansel D - PLoS Comput. Biol. (2007)

Bottom Line: In contrast, when the Na(+) window current is large, the neuron always fires tonically.We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current.We report experimental results from intracellular recordings supporting this prediction.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology, Ben-Gurion University, Be'er-Sheva, Israel. golomb@bgu.ac.il

ABSTRACT
Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na(+), delayed-rectifier K(+), and slowly inactivating d-type K(+) conductances. The model is analyzed using nonlinear dynamical system theory. For small Na(+) window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, gd, and it is delayed for larger gd. As gd further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na(+) window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.

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Fast–Slow Analysis of Neurons with Small Na+ Window Current Exhibiting a Depolarized Rest Potential or StutteringFast–slow analysis is described for θm = −24 mV.(A,B) Parameters are: gd = 0.39 mS/cm2, Iapp = 2.9 μA/cm2. (Except for Iapp, parameters are as in Figures 5A–5C and 2C. The neuron is quiescent at steady state.)(C,D) The parameters gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2 are as in Figure 2D, and the neuron stutters at steady state. For each case, we plot the bifurcation diagrams of the fast subsystem in the V-b space (A,C), and the functions b∞(VFP(b)) for fixed points and F(b) for limit cycles (Equation 2) as functions of b (B,D). Symbols are as in Figure 5.
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pcbi-0030156-g006: Fast–Slow Analysis of Neurons with Small Na+ Window Current Exhibiting a Depolarized Rest Potential or StutteringFast–slow analysis is described for θm = −24 mV.(A,B) Parameters are: gd = 0.39 mS/cm2, Iapp = 2.9 μA/cm2. (Except for Iapp, parameters are as in Figures 5A–5C and 2C. The neuron is quiescent at steady state.)(C,D) The parameters gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2 are as in Figure 2D, and the neuron stutters at steady state. For each case, we plot the bifurcation diagrams of the fast subsystem in the V-b space (A,C), and the functions b∞(VFP(b)) for fixed points and F(b) for limit cycles (Equation 2) as functions of b (B,D). Symbols are as in Figure 5.

Mentions: We now turn to the dynamics of the full system. Let us assume that the neuron is at rest (Iapp = 0) and that the inactivation variable b is equal to brest (brest = 0.5 in the example of Figures 2C and 5A). At time t = 0, a current step is applied and Iapp is raised abruptly (e.g., 3.35 μA/cm2 in Figure 5A–5C and 1.25 μA/cm2 in Figure 5D–5F). The evolution of the membrane potential of the neuron, V, right after the step onset is driven first by the dynamics of the fast subsystem with b ∼ brest (Figures 5A and 5D). The membrane depolarizes rapidly, the current Id starts to inactivate, and the variable b decreases. During this process, which occurs on the slow time scale of τb, the state of the neuron follows the fixed point of the fast subsystem adiabatically, for the applied current Iapp. In particular, the membrane potential of the neuron is V(t) ∼ VFP(b(t)), and the slow variable b continues to decrease as long as b∞(VFP(b(t))) < b (Equation 17). If the equation b∞(VFP(b)) = b has a solution, b*, for which the fixed point of the fast subsystem is stable, b stops evolving when it reaches that value. In that case, the state of the neuron converges to a stable fixed point where it does not fire action potentials. This situation, which happens when Iapp is small, is depicted in Figure 6A and 6B. The value b* decreases with Iapp. The largest value of Iapp for which the fixed point is stable is determined by the equationor by the equation b∞(VFP(bSN; Iapp)) = bSN for small and large window INa, respectively. For larger values of Iapp, a solution of the equation b∞(VFP(b)) = b exists only for a b value for which the fixed point of the fast subsystem is unstable (Figure 5C and 5F). As a result, b keeps decreasing until it crosses the bifurcation of the fast subsystem (b = bHopf or b = bSN). When this happens, V starts to diverge from VFP(b) and the neuron fires action potentials. The patterns of firing following the delay can also be assessed using the fast–slow analysis as we will explain below.


Mechanisms of firing patterns in fast-spiking cortical interneurons.

Golomb D, Donner K, Shacham L, Shlosberg D, Amitai Y, Hansel D - PLoS Comput. Biol. (2007)

Fast–Slow Analysis of Neurons with Small Na+ Window Current Exhibiting a Depolarized Rest Potential or StutteringFast–slow analysis is described for θm = −24 mV.(A,B) Parameters are: gd = 0.39 mS/cm2, Iapp = 2.9 μA/cm2. (Except for Iapp, parameters are as in Figures 5A–5C and 2C. The neuron is quiescent at steady state.)(C,D) The parameters gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2 are as in Figure 2D, and the neuron stutters at steady state. For each case, we plot the bifurcation diagrams of the fast subsystem in the V-b space (A,C), and the functions b∞(VFP(b)) for fixed points and F(b) for limit cycles (Equation 2) as functions of b (B,D). Symbols are as in Figure 5.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC1941757&req=5

pcbi-0030156-g006: Fast–Slow Analysis of Neurons with Small Na+ Window Current Exhibiting a Depolarized Rest Potential or StutteringFast–slow analysis is described for θm = −24 mV.(A,B) Parameters are: gd = 0.39 mS/cm2, Iapp = 2.9 μA/cm2. (Except for Iapp, parameters are as in Figures 5A–5C and 2C. The neuron is quiescent at steady state.)(C,D) The parameters gd = 1.8 mS/cm2 and Iapp = 4.2 μA/cm2 are as in Figure 2D, and the neuron stutters at steady state. For each case, we plot the bifurcation diagrams of the fast subsystem in the V-b space (A,C), and the functions b∞(VFP(b)) for fixed points and F(b) for limit cycles (Equation 2) as functions of b (B,D). Symbols are as in Figure 5.
Mentions: We now turn to the dynamics of the full system. Let us assume that the neuron is at rest (Iapp = 0) and that the inactivation variable b is equal to brest (brest = 0.5 in the example of Figures 2C and 5A). At time t = 0, a current step is applied and Iapp is raised abruptly (e.g., 3.35 μA/cm2 in Figure 5A–5C and 1.25 μA/cm2 in Figure 5D–5F). The evolution of the membrane potential of the neuron, V, right after the step onset is driven first by the dynamics of the fast subsystem with b ∼ brest (Figures 5A and 5D). The membrane depolarizes rapidly, the current Id starts to inactivate, and the variable b decreases. During this process, which occurs on the slow time scale of τb, the state of the neuron follows the fixed point of the fast subsystem adiabatically, for the applied current Iapp. In particular, the membrane potential of the neuron is V(t) ∼ VFP(b(t)), and the slow variable b continues to decrease as long as b∞(VFP(b(t))) < b (Equation 17). If the equation b∞(VFP(b)) = b has a solution, b*, for which the fixed point of the fast subsystem is stable, b stops evolving when it reaches that value. In that case, the state of the neuron converges to a stable fixed point where it does not fire action potentials. This situation, which happens when Iapp is small, is depicted in Figure 6A and 6B. The value b* decreases with Iapp. The largest value of Iapp for which the fixed point is stable is determined by the equationor by the equation b∞(VFP(bSN; Iapp)) = bSN for small and large window INa, respectively. For larger values of Iapp, a solution of the equation b∞(VFP(b)) = b exists only for a b value for which the fixed point of the fast subsystem is unstable (Figure 5C and 5F). As a result, b keeps decreasing until it crosses the bifurcation of the fast subsystem (b = bHopf or b = bSN). When this happens, V starts to diverge from VFP(b) and the neuron fires action potentials. The patterns of firing following the delay can also be assessed using the fast–slow analysis as we will explain below.

Bottom Line: In contrast, when the Na(+) window current is large, the neuron always fires tonically.We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current.We report experimental results from intracellular recordings supporting this prediction.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology, Ben-Gurion University, Be'er-Sheva, Israel. golomb@bgu.ac.il

ABSTRACT
Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na(+), delayed-rectifier K(+), and slowly inactivating d-type K(+) conductances. The model is analyzed using nonlinear dynamical system theory. For small Na(+) window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, gd, and it is delayed for larger gd. As gd further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na(+) window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.

Show MeSH
Related in: MedlinePlus