Our model of FS cells is based on that of [3 (
link),28 (
link)], with several modifications based on voltage clamp data. The current balance equation is
where
V is the membrane potential of the neuron,
C = 1μF/cm
2 is the membrane capacitance, and the parameters of the leak current are
gL = 0.25 mS/cm
2 and
VL = −70 mV. The external current injected into the neuron is denoted by
Iapp.
The Na
+ current
INa is given by:
where the gating variables,
h and
m, follow:
The parameters are:
gNa = 112.5 mS/cm
2,
VNa = 50 mV,
σm = 11.5 mV,
θh = −58.3 mV,
σh = −6.7 mV,
θth = −60 mV,
σth = −12 mV [14 (
link)]. In this work, we study the effect of the strength of the Na
+ window current, controlled by the parameter
θm, on the dynamics of the neuron.
The delayed rectifier K
+ current
IKdr is of the Kv3.1–Kv3.2 type. It is responsible for the brief duration of the spike, about 0.5 ms [2 (
link),48 (
link)], and for the high firing frequency [3 (
link),49 (
link)]. It is given by:
with:
All the parameters of the delayed rectifier current are fixed:
gKdr = 225 mS/cm
2,
VK = −90 mV,
θn = −12.4 mV,
σn = 6.8 mV,
θtn = −27 mV,
σth = −15 mV [50 (
link)].
The K
+ current
Id incorporated in the model [10 (
link),11 (
link)] has fast activation and slow inactivation. It is defined by:
Throughout the paper, all the parameters of the d-current but
gd are fixed:
θa = −50 mV,
σa = 20 mV,
τa = 2 ms,
θb = −70 mV,
σb = −6 mV,
τb = 150 ms [51 (
link),52 (
link)]. The parameter
gd is varied to study the effect of the strength of this current.
Finally, to study the effect of noise in the external input on the firing pattern of the neuron, we add an additional external input,
Inoise, of the form:
where
ξ(
t) is a Gaussian white noise with an average 0 and a unit variance, and
D has the units of μA
2 × ms/cm
4 .
Numerical methods. Simulations were performed using the fourth-order Runge-Kutta method with a time step of 0.01 ms implemented as a C program or within the software package XPPAUT [53 ], which was used also for computing bifurcation diagrams.
Delay. The delay duration
tdelay is defined to be the time from the onset of current injection, or, if the neuron fires transient 1–3 spikes, from the last transient spike to the first spike of the sustained firing. We define that the neuron shows a delay if
tdelay is at least twice as large as the inter-spike interval during steady-state spiking
tISI, or if it is larger than both 100 ms and 1.2
tISI.
Fourier spectrum. Discrete Fourier transforms of subthreshold oscillations were calculated numerically over a time window of
TFT ending
TBS = 5 ms before the first spike of the steady-state firing. The absolute values of the Fourier components were averaged over
nR repetitions of the same stimulus. Parameters for
Figure 9 are:
TFT = 120 ms,
nR = 20. Parameters for
Figure 12 are:
TFT = 90 ms,
nR = 5 (A),
TFT = 120 ms,
nR = 13 (B),
TFT = 120 ms,
nR = 11 (C).
Golomb D., Donner K., Shacham L., Shlosberg D., Amitai Y, & Hansel D. (2007). Mechanisms of Firing Patterns in Fast-Spiking Cortical Interneurons. PLoS Computational Biology, 3(8), e156.