Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 433

In Figure 15.1 , sample paths of V ( t ) in the presence of the original input current

I obtained by numerical integration of Equation (15.1) are compared with sample

paths in the presence of the effective input I

( t )

( t ) , obtained using Equation (15.16). As

illustrated in Figure 15.1, t re f =

( )

2 ms after emitting a spike, V t begins to integrate

( )

its inputs again starting from V r until it reaches V th . reaches

V th is called the 'first-passage time' (denoted by T fp ). Taking the refractory period
into account, the whole interval between consecutive spikes is called the inter-spike
interval (ISI). Therefore, the statistics of ISIs can be analyzed using the theory of
first-passage times of the Ornstein-Uhlenbeck process [97, 117].

The first time V t

15.2.4 Computation of the mean firing rate and CV

The Fokker-Planck Equation (15.9) can be rewritten as a continuity equation by
defining

s V
2t m

) ≡− (

V − V ss )

t m

∂
∂ V

S ( V , t | V 0 , t 0 r ( V , t | V 0 , t 0 ) − r ( V , t | V 0 , t 0 )] , (15.17)

so that Equation (15.9) becomes

∂
∂ t

∂
∂ V S

r ( V , t | V 0 , t 0 )= − ( V , t | V 0 , t 0 ) . (15.18)

Thus, S is the flux of probability (or probability current) crossing V at

time t . To proceed, a set of boundary conditions on t and V has to be specified for
r

( V , t | V 0 , t 0 )

. First one notices that, if a threshold exists, then the voltage can only
be below threshold and can only cross it from below (the threshold is said to be
an absorbing barrier). The probability current at threshold gives, by definition, the
average firing rate of the cell. Since r

( V , t | V 0 , t 0 )

( V > V th , t | V 0 , t 0 )= 0, the probability density

must be zero at V V th and

so would be the firing rate according to Equation (15.17). Therefore, we have the
following boundary conditions

= V th , otherwise the derivative would be infinite at V =

∂
∂ V

2n ( t ) t m

r ( V th , t | V 0 , t 0 )= 0 and r ( V th , t | V 0 , t 0 )= − , (15.19)

s V

for all t . The conditions at V = − • ensure that the probability density vanishes fast

enough to be integrable, i.e.,

lim r ( V , t | V 0 , t 0 )= 0 and lim

• V r

( V , t | V 0 , t 0 )= 0 . (15.20)

V →− • V →−

Since the threshold is an absorbing boundary, a finite probability mass is con-
stantly leaving the interval ( − • , V th ) . Under this condition, there is no stationary

distribution for the voltage, i.e., •. In order to study the

steady-state of the process, one can keep track of the probability mass leaving the
integration interval at t , and re-inject it at the reset potential at t

r ( V , t | V 0 , t 0 ) → 0as t →

+ t re f . This injection