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

2112 G. La Camera, A. Rauch, H.-R. Lüscher, W. Senn, and S. Fusi
firing rate [Hz]
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CELL A CELL B
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m [pA]
AHP
AT
Figure 5: Best fits of different models to the rate functions of two rat pyramidal
cells (see appendix B). Models: LIF neuron with AHP adaptation (AHP) and with
an adapting threshold (AT). Theoretical curves (full lines) and experimental
points (dots) plotted as in Figure 1A. Best-fit parameters are: Cell A: AHP:
τ r = 6.6 ms, τ = 27.1 ms, C = 260 pF, V r = 1.7 mV,α = 5.1 pA·s, P = 0.32;
AT: τ r = 2.2 ms, τ = 28.8 ms, C = 270 pF, V r = 14 mV, β = 0.58 mV·s,
P = 0.38. Cell B: AHP: τ r = 19.8 ms, τ = 41.1 ms, C = 440 pF, V r =−2mV,
α = 2.8pA·s, P = 0.85; AT: τ r = 6.8 ms, τ = 40.1 ms, C = 430 pF, V r =−12.8mV,
β = 0.30 mV·s, P = 0.80. In all the fits, the threshold (θ 0 in AT) was kept fixed
to 20 mV. P equals the probability that χ 2 is larger than the observed minimum
χmin 2 . The fit was accepted whenever P > 0.01. Amplitude of the fluctuating
current: cell A: s = 0, 200 and 400 pA; cell B: s = 50, 200, 400 and 500 pA.
is required to bend the response of the model with AHP, otherwise linear
in that region.
3 Adapting Response to Time-Dependent Inputs
The stationary response function can be used also to predict the timevarying
activity of a population of adapting neurons, as shown in this
section. Consider an input spike train of time-varying frequency ν x (t), targeting
each cell of the population through x-receptor mediated channels.
Each spike contributes a postsynaptic current of the form ḡ x e −t/τ x
, where
ḡ x is the peak conductance of the channels. In the diffusion approximation,
such an input I x is an Ornstein-Uhlenbeck (OU) process with average
¯m x = ḡ x ν x (t)τ x , variance ¯s 2 x (t) = (1/2)ḡ2 x ν x(t)τ x , and correlation length τ x :
√
dI x =− I x − ¯m x 2dt
dt + ¯s x ξ t , (3.1)
τ x
τ x
where ξ t is the unitary gaussian process defined in section A.1.