Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 436

A B C

80

1 1

60

0.75 0.75

40

0.5 0.5

20

0.25 0.25

0

0.32 0.36 0.4 0.44 0.48 0.32 0.36 0.4 0.44 0.48 0 25 50 75

m

C (nA)

m

C (nA)

Firing Rate (Hz)

Figure 15.2
Firing rate (A) and CV (B) of the LIF neuron as a function of the mean current m C for
three values of the effective standard deviation in the voltage s V 1 mV (solid),

1 mV (dashed) and 4 mV (dot-dashed). (C) CV as a function of the mean firing
rate when m C is varied as in the three curves in A-B . The parameters of the cell are
C m =

= 0 .

0 . 2nF, g L = 20 nS (t m = 10 ms), t re f = 2ms, V L = − 70 mV, V th = − 50 mV

and V r = − 60 mV.

/ √ p

x

0 e u 2 du .
Once the firing rate is known, the following recursive relationship between the
moments of the first-passage time distribution of the Ornstein-Uhlenbeck process
(see e.g., [117]) can be used to find

where erf ( x )= 2

T fp > <

d 2 T pf >

dx 2

T k fp >
dx

< d <

s V
2

T k − 1
fp

+( V ss − x ) = − k < >, (15.27)

where x = V r . Given that < T fp > = 1 / n in Equation (15.26), the CV of the ISI is

given by [19]

2pn 2 V th − V ss dxe x 2

T fp > − <

2

≡ <

T fp >

x

s V
V r − V ss
s
V

• dye y 2

CV 2

= ( 1 + erf ( y )) . (15.28)

<

T fp >

2

−

In Figures 15.2A-B we plot the mean firing rate and CV of the LIF neuron as given
by Equations (15.26,15.28), as a function of the mean current m C for various values
of the effective standard deviation in the voltage s V . The firing rate (Figure 15.2A)
is a monotonic increasing function of the average current. Qualitatively, it starts
to rise when the average current comes within a standard deviation of the current
threshold, defined as I th ≡ , and shown as a vertical dashed line in Figure

15.2A. It increases supra-linearly with m C for sub-threshold mean currents, and sub-
linearly when the mean current is above threshold, eventually saturating at 1

g L ( V th − V L )

/ t re f .

Therefore, for a wide range of values of s V , m C ∼ I th is close to the point where the

curvature of n ( m C ) changes sign. The deterministic current threshold I th also marks