Exact differential equation population dynamics for integrate-and-fire neurons
Authors: Julian Eggert, Berthold Bäuml
Published in: Advances in Neural Information Processing Systems 14  (NIPS 2001) (Main Conference Track), 2001

Abstract: In  our  previous  work,  integral  equation  formulations  for 
Mesoscopical,  mathematical  descriptions  of  dynamics  of  popula(cid:173) tions of spiking neurons  are getting increasingly important for  the  understanding  of large-scale  processes  in  the  brain  using  simula(cid:173) tions.  population dynamics  have been derived for  a  special  type of spik(cid:173) ing neurons.  For  Integrate- and- Fire type neurons, these formula(cid:173) tions  were  only  approximately  correct.  Here,  we  derive  a  math(cid:173) ematically  compact,  exact  population  dynamics  formulation  for  Integrate- and- Fire type  neurons.  It can  be  shown  quantitatively  in  simulations that the numerical  correspondence with  microscop(cid:173) ically modeled neuronal populations is  excellent. 
1 
Introduction and  motivation 
The goal of the population dynamics approach is to model the time course of the col(cid:173) lective activity of entire populations of functionally and dynamically similar neurons  in a compact way,  using a  higher descriptionallevel than that of single neurons and  spikes.  The usual observable at the level of neuronal populations is  the population(cid:173) averaged instantaneous firing  rate  A(t),  with  A(t)6.t being the number of neurons  in the population that release a spike in an interval [t, t+6.t).  Population dynamics  are formulated  in  such  a  way, that they match quantitatively the time  course  of a  given  A(t), either gained experimentally or by microscopical, detailed simulation. 
At  least  three  main  reasons  can  be  formulated  which  underline  the  importance  of  the  population  dynamics  approach  for  computational  neuroscience.  First,  it  enables the simulation of extensive networks involving a massive number of neurons 
and connections, which is  typically the case when dealing with biologically realistic  functional  models  that go  beyond  the single neuron level.  Second,  it increases the  analytical understanding of large-scale neuronal dynamics, opening the way towards  better control and predictive capabilities when dealing with large networks.  Third,  it enables  a  systematic  embedding  of the  numerous  neuronal  models  operating at  different  descriptional scales into a  generalized theoretic framework, explaining the  relationships,  dependencies  and derivations of the respective models. 
Early efforts on population dynamics approaches date back as early as 1972, to the  work of Wilson  and  Cowan  [8]  and  Knight  [4],  which  laid  the  basis  for  all  current  population-averaged graded-response  models  (see  e.g.  [6]  for  modeling  work  using  these models).  More recently, population-based approaches for spiking neurons were  developed,  mainly by Gerstner [3, 2]  and Knight  [5].  In our own previous work  [1],  we  have  developed  a  theoretical framework  which enables to systematize and sim(cid:173) ulate  a  wide  range  of models  for  population-based  dynamics.  It  was  shown  that  the equations of the framework produce results that agree quantitatively well  with  detailed  simulations  using  spiking  neurons,  so  that  they  can  be  used  for  realistic  simulations  involving  networks  with  large  numbers  of spiking  neurons.  Neverthe(cid:173) less,  for  neuronal populations composed of Integrate-and-Fire (I&F)  neurons, this  framework was only correct in an approximation.  In this paper, we  derive the exact  population  dynamics  formulation  for  I&F  neurons.  This  is  achieved  by  reducing  the  I&F  population  dynamics  to  a  point  process  and  by  taking  advantage  of the  particular properties of I&F neurons. 
2  Background:  Integrate-and-Fire dynamics 
2.1  Differential form 
We  start  with  the  standard Integrate- and- Fire  (I&F)  model  in  form  of the  well(cid:173) known differential equation [7] 
(1) 
which  describes  the  dynamics  of the  membrane  potential  Vi  of a  neuron  i  that  is  modeled  as  a  single  compartment  with  RC  circuit  characteristics.  The  membrane  relaxation time is in this case T  =  RC with R being the membrane resistance and C  the  membrane  capacitance.  The resting  potential  v R est  is  the  stationary potential  that is  approached in  the no-input  case.  The input arriving from  other neurons  is  described in form  of a  current ji. 
In  addition  to  eq.  (1),  which  describes  the  integrate  part  of the  I&F  model,  the  neuronal  dynamics  are  completed  by  a  nonlinear  step.  Every  time  the  membrane  potential Vi  reaches a fixed  threshold ()  from  below,  Vi  is  lowered by a fixed  amount  Ll  > 0,  and from the new value of the membrane potential integration according to  eq.  (1)  starts again. 
if Vi(t)  =  ()  (from  below)  . 
(2) 
At  the  same  time,  it  is  said  that  the  release  of a  spike  occurred  (i.e.,  the  neuron  fired),  and  the  time  ti  =  t  of this  singular  event  is  stored.  Here  ti  indicates  the  time of the  most recent spike.  Storing all the last firing  times, we  gain the sequence  of spikes  {t{}  (spike ordering index  j, neuronal index i). 
2.2