Minimal Models of Adapted Neuronal Response to In Vivo–Like ...

Adapting Rate Models 2113
Table 1: Summary of the Results of the Fit of the LIF Neuron to the Experimental
Rate Functions of 26 Rat Neocortical Pyramidal Cells.
AHP
AT
N 14 13
α [pA · s], β [mV · s] 4.3 ± 2.2 0.29 ± 0.13
τ r [ms] 9.0 ± 6.5 3.0 ± 4.0
τ [ms] 33.2 ± 9.4 32.5 ± 9.2
C [nF] 0.50 ± 0.18 0.50 ± 0.18
V r [mV] 0.1 ± 11.2 1.1 ± 13.7
P 0.40 ± 0.30 0.33 ± 0.29
Notes: N is the number of fitted cells that required an
adaptation parameter (α, orβ) > 0. Two cells could
be fitted without adaptation and were not included
in the analysis. The parameters (left-most column)
are defined in section 2.1 and their best-fit values are
reported as average ± SD. The threshold for spike
emission was held fixed to 20 mV. P is the probability
P[χ 2 >χmin 2 ] across fitted cells requiring adaptation.
A fit was accepted whenever P > 0.01. The threshold
for spike emission was held fixed to 20 mV. AT:
adapting threshold model.
The population activity of noninteracting neurons is well predicted by
f (t) = (m x , s 2 x ), where is the stationary response function, and m x, s 2 x
are the time-varying average and variance of I x (see, e.g., Renart, Brunel, &
Wang, 2003). These evolve according to the first-order dynamics (ẏ ≡ dy/dt),
τ x ṁ x =−(m x − ¯m x ), (3.2)
and analogously for s 2 x , with τ x replaced by τ x /2 (e.g., Gardiner, 1985). We
now include adaptation in the following way:
f = (m x − I ahp , s 2 x )
τ N İ ahp =−I ahp + αf, (3.3)
where I ahp is the AHP current, which follows the instantaneous output rate
with time constant τ N . Note that for a stationary stimulus, that is, ν x constant,
after a transient of the order of max{τ x ,τ N }, one recovers the stationary
model, equation 2.2, with m = ¯m x , s = ¯s x .
In the case of several independent components, they follow their own
synaptic dynamics and sum up in the argument of the response function to
give the time-varying firing rate:
( ∑
f = m x − I ahp , ∑ )
s 2 x .
x
x