Information through a Spiking Neuron
Authors: Anthony M. Zador Charles F. Stevens
Published in: Advances in Neural Information Processing Systems 8  (NIPS 1995) (Main Conference Track), 1995

Abstract: While  it  is  generally  agreed  that  neurons  transmit  information  about their synaptic inputs through spike trains, the code by which  this  information is  transmitted  is  not  well  understood.  An  upper  bound  on  the  information  encoded  is  obtained  by  hypothesizing  that the precise  timing of each spike conveys information.  Here  we  develop  a  general  approach  to quantifying the information carried  by  spike  trains  under  this  hypothesis,  and  apply  it  to  the  leaky  integrate-and-fire  (IF)  model  of  neuronal  dynamics.  We  formu(cid:173) late  the  problem  in  terms  of the  probability  distribution  peT)  of  interspike  intervals  (ISIs),  assuming that  spikes  are  detected  with  arbitrary  but finite  temporal resolution .  In  the  absence  of added  noise,  all  the  variability in  the  ISIs  could encode  information, and  the  information rate  is  simply the entropy of the lSI  distribution,  H (T)  =  (-p(T) log2 p(T)},  times  the  spike  rate.  H (T)  thus  pro(cid:173) vides  an  exact  expression  for  the  information rate.  The  methods  developed  here  can be  used  to determine experimentally the infor(cid:173) mation carried  by  spike  trains,  even  when  the  lower  bound of the  information rate  provided  by  the  stimulus reconstruction  method  is  not  tight.  In  a  preliminary series  of experiments,  we  have  used  these  methods  to estimate information rates of hippocampal neu(cid:173) rons  in  slice  in  response  to somatic  current  injection.  These  pilot  experiments suggest  information rates as  high as  6.3  bits/spike. 
1 
Information rate of spike trains 
Cortical neurons use  spike  trains  to communicate with other  neurons.  The output  of each  neuron  is  a  stochastic function of its input from  the  other neurons.  It is  of  interest  to know  how  much each  neuron  is  telling other neurons about its inputs. 
How  much information does  the spike train provide about a signal?  Consider noise  net)  added  to  a  signal  set)  to  produce  some  total  input  yet)  =  set)  + net).  This  is  then  passed  through  a  (possibly  stochastic)  functional  F  to produce  the  output  spike  train  F[y(t)]  --+  z(t).  We  assume  that  all the  information contained  in  the  spike  train  can  be  represented  by  the  list  of spike  times;  that  is,  there  is  no extra  information contained in  properties  such  as spike  height or  width.  Note,  however,  that many characteristics of the spike  train such  as the mean or instantaneous rate 
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can be  derived  from  this  representation;  if such  a  derivative  property  turns  out  to  be  the relevant one,  then  this formulation can be specialized  appropriately. 
We  will  be  interested,  then,  in  the  mutual  information 1(S(t); Z(t»  between  the  input signal ensemble S(t) and the output spike train ensemble Z(t) . This is defined  in  terms  of the  entropy  H(S)  of the  signal,  the  entropy  H(Z)  of the  spike  train,  and  their joint entropy  H(S, Z), 
1(S; Z) =  H(S) + H(Z) - H(S, Z). 
(1)  Note that the mutual information is  symmetric, 1(S; Z) =  1(Z; S),  since  the joint  entropy  H(S, Z)  =  H(Z, S).  Note  also  that  if the  signal  S(t)  and  the  spike  train  Z(t)  are  completely independent,  then  the mutual information is  0,  since  the joint  entropy is just the sum of the individual entropies H(S, Z) = H(S) + H(Z).  This is  completely in lin'e  with our intuition, since  in  this case  the  spike  train  can  provide  no  information about the signal. 
Information estimation through stimulus reconstruction 
1.1  Bialek  and  colleagues  (Bialek  et  al.,  1991)  have  used  the  reconstruction  method  to  obtain  a  strict  lower  bound  on  the  mutual information in  an experimental set(cid:173) ting.  This method is  based on  an expression  mathematically equivalent  to eq.  (1)  involving the conditional entropy  H(SIZ) of the signal given the spike train,