Phase‐response curves of ion channel gating kinetics

Abstract

Reduced models of single-neuron dynamics are widely used to characterize neuronal properties such as stimulus encoding and a cell's contribution to network function. Low-dimensional phase oscillator models based on phase-response curves, in particular, permit to capture the essence of a neuron's input-output dynamics at the level of spike times from high-dimensional conductance-based neuron models. In principle, these phase models allow for a direct translation of perturbations in the state variables (such as changes in specific ion channel conductances and kinetics) to shifts in the timing of spikes. Previous mathematical work, however, has predominantly focused on voltage and current perturbations and less on perturbations occurring in any of the involved kinetic equations of ionic channels. Here, we provide the means to mathematically relate properties in these variables to the phase-dependent voltage dynamics. Specifically, we derive the vector of phase-response curves near two different onset bifurcations: the physiologically prominent saddle node on invariant cycle bifurcation as well as the saddle-node loop bifurcation. We demonstrate that, locally around the saddle node, the tangent space of the isochrons is spanned by the strongly stable manifold of the saddle node. This finding enables us to quantitatively relate perturbations in the gating kinetics of ion channels to the resulting changes in spike timing. These results lay the methodological foundation for future quantitative analyses of the impact of specific biophysical channel properties on spike timing, including, but not limited to, channel noise. KEYWORDSconductance-based neuron model, gating kinetics, homoclinic orbits to a saddle node, isochrons, phase-response curves, saddle-node loop, saddle node on invariant cycle, semistable manifold, strongly stable manifold 8844

A dynamic clamp protocol to artificially modify cell capacitance

“…How the form of the f-I curve depends on capacitance can be analytically calculated for a single-compartment conductance-based neuron model undergoing a saddle-node on a limit cycle bifurcation at spiking onset like the WB model considered here ( Izhikevich, 2006 ), pp. 162–168; ( Schleimer and Schreiber, 2018 ). In this case, the time between two spikes is dominated by the slow traversal T 2 of the saddle node, which close to threshold is multiple times longer than the brief duration T 1 of the spike and can be derived by considering solely local dynamics …”

“…162–163. In principle, and can be calculated for arbitrary gating kinetics by projecting the dynamics on the center manifold ( Schleimer and Schreiber, 2018 ). For the assumption of small gating time constants, however, they can be expressed in simpler terms using the steady state I-V relation of the neuron divided by its membrane capacitance taking the form and , where the saddle node voltage and current are given by the equations In summary, both and are proportional to and the rheobase current is independent of capacitance, thus confirming the expected scaling.…”

Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

“…Equation (14) and other relationships between the state and the dynamics in directions transverse to a stable periodic orbit are particularly important in the development of second order accurate reduction strategies. These issues have been explored previously and the interested reader is referred to [8], [22], [21] for a detailed discussion. Substituting Eq.…”

“…In an analogous strategy to the one used in the previous section, one can use the continuity assumption on θ and ψ j , the relation ∂ψj ∂x • F(x) = κ j ψ j , and Eq. (22) to derive the jump conditions across Π:…”

“…For instance [8] uses such an expansion to obtain the frequency change of an oscillator in response to parameter perturbations. Additionally, [21] and [22] use higher order corrections to characterize neural spiking statistics in response to ion channel fluctuations wherein the notion of radial susceptibilities is analogous to the IRCs considered in this work. The transformation of the dynamical system from Eq.…”