Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 475

are called the infinitesimal moments of the process. The intuitive nature of the diffu-
sion approximation becomes now clear: the smaller J E , I , the fewer the terms needed
to express r

J E , I ,

as a Taylor series expansion arround V ,andthefewer
the terms one has to maintain in the infinite-order Equation (15.103) to give an accu-
rate description of the process.

( V ∓ t | V 0 , t 0 )

Appendix 2: Stability of the steady-state solutions
for r ss ( V )
The function r ss ( is the solution of the stationary Fokker-Planck Equation (15.23)

with the appropriate boundary conditions. To assess the dynamical stability of this
solution, one has to use the general Fokker-Planck Equation (15.9) to test the effect
of small perturbations on the steady-state distribution. We briefly mention the logic
of this procedure.
For the sake of simplicity, we only discuss here a simple situation in which synapses
are instantaneous, with a latency t l ms after the pre-synaptic spike time. In this case,
the dynamical counterparts to Equations (15.63, 15.68) are

V )

s ext
2

∂ 2 r
∂ V 2 −

∂r
∂ t =

∂
∂ V [(

J t m n

t m V − m ext − ( t − t l )) r ] , (15.106)

in the weakly coupled, fully connected case (no noise in recurrent inputs), and

s ext +

) C J 2 t m n

( t − d

∂ 2 r
∂ V 2

∂r
∂ t =

t m

2

∂
∂ V [(

C J t m n − V − m ext − ( t − t l )) r ] , (15.107)

in the strongly coupled, sparsely connected case (noise in recurrent inputs). In both
cases the boundary conditions are given by Equations (15.19,15.20).
The stationary solution to Equation (15.106) (resp. 15.107) is Equation (15.63)
(resp. 15.68). To study their stability, a linear stability analysis must be performed.
It consists in looking for solutions to Equations (15.106, 15.107) of the form

r ( V , t | V 0 , t 0 )= r ss ( V )+ dr ( V , l ) exp ( l t ) (15.108)

n ( t )= n ss + dn ( l ) exp ( l t ) , (15.109)

where r ss , n ss correspond to the stationary solution, drand dnare small deviations
around the stationary solution that evolves in time with the (complex) growth rate
l. Upon inserting Equation (15.109) in Equations (15.106) or (15.107), and keeping
the term first order in drand dn, an equation results for the possible growth rates l.
Solutions with Re

( l ) > 0 indicate that the stationary state is unstable. Instabilities