Poisson Model of Spike Generation

1 Poisson Model of Spike GenerationProfessor David HeegerSeptember 5, 2000In the cortex, the timing of successive action potentials is highly irregular. The interpretationof this irregularity has led to two divergent views of cortical organization. On the one hand, theirregularity might arise from stochastic forces. If so, the irregular interspike interval reflects arandom process and implies that an instantaneous estimate of the Spike rate can be obtained byaveraging the pooled responses of many individual neurons. In keeping with this theory, onewould expect that the precise timing of individual spikes conveys little information. Alternatively,the irregular ISI may result from precise coincidences of presynaptic events. In this scenario, it ispostulated that the timing of spikes, their intervals and patterns can convey information.

2 Accordingto this view, the irregularity of the ISI reflects a rich bandwidth for information this handout, we take the former point of view, that the irregular interspike interval reflectsa random process. We assume that the Generation of each Spike depends only on an underlyingcontinuous/analog driving signal,r(t), that we will refer to as the instantaneous firing rate. Itfollows that the Generation of each Spike is independent of all the other spikes, hence we refer tothis as theindependent Spike the independent Spike hypothesis were true, then the Spike train would be completely de-scribed a particular kind of random process called aPoisson process. Note that even though aPoisson Spike train is generated by a random process, some stimuli could still evoke spikes veryreliably by forcing the instantaneous firing rate to be very large at particular moments in time sothat the probability of firing would then be arbitrarily close to features of neuronal firing, however, violate the independent Spike hypothesis.

3 Fol-lowing the Generation of an action potential, there is an interval of time known as the absoluterefractory period during which the neuron can not fire another Spike . For a longer interval knownas the relative refractory period, the likelihood of a Spike being fired is much reduced. Bursting isanother non- Poisson feature of neuronal spiking. Some neurons fire action potentials is clusters orbursts, and these tend to be poorly described a purely Poisson Spike - Generation process. Below, Ipresent ways of extending the Poisson Model to account for refractoriness and and Random ProcessesArandom variableis a number assigned to every outcome of an experiment. This could be theoutcome of the roll of a die, or the number of action potentials generated by a visual neuron duringa 1 sec stimulus presentation.

4 The probability of getting each possible outcome is characterizedby aprobability density function. For a fair die, there is a 1/6 probability of getting each possi-ble outcome. The familiar bell-shaped curve of the normal distribution is another example of aprobability density integral of a probability density function is called thecumulative distribution distributions characterize the probability of getting an outcome less than or equal tosome specified value. For example, there is a 1/2 probability of getting a roll less than or equal to3 on a fair random process is a rule for assigning a functionx(t)to every outcome of an example, the voltage trace recorded from an intracellular electrode during a 1 sec stimuluspresentation might be considered a random Firing RateDefine (t), theneural response function, to be a bunch of impulses, one for each action potential: (t)=kXi=1 (t ti);wherekis the total number of spikes in the Spike train, andtiare the times that each Spike unit impulse signal is defined as: (t)=(1ift=00otherwise;such that the integral of (t)is one:Z1 1 (t)=1:We would like to think of the neural response function as a random process.)

5 The neural responsefunction is completely equivalent to a list of the Spike times in the Spike train. Nevertheless, it isuseful for re-expressing sums over spikes as integrals over time. For example, we can write thespike count, the number of spikes fired between timest1andt2as the integral:n=Zt2t1 (t)dt;because each Spike contributes1to the instantaneous firing rate ( , of a sensory neuron) can now be formally defined to be theexpectation of the neural response function, averaged over an infinite number of repeats ( , of2the same stimulus presentation):r(t)=h (t)i:In practice, of course, you can not run an infinite number of trials. The function you get byaveraging over a finite number of trials, is an estimate of the instantaneous firing rate:rM(t)=1 MMXj=1 j(t);whereMis the number of trials and j(t)is the neural response function for each trial.

6 This, ofcourse, is not a continuous function because it is just a sum of functions. You get a smoothfunction only in the formal limit with an infinite number of trials. Typically, when working withreal data, you would blurrMto make it smooth. We do not have to worry about that in this classbecause the theoretical/computational neuroscientist has the luxury of being able to just make upa continuous function,r(t).The average Spike count can then be defined from the instantaneous firing rate:hni=Zt2t1r(t)dt:(1)This is equivalent, of course, to counting the spikesnjin each of a very large ( , infinite) numberof repeated trials, and then averaging those Spike counts across the sufficiently small intervals, whent2=t+ t=2andt1=t t=2, the average Spike countcan be approximated byhni=r(t) t.

7 Furthermore, tcan be reduced until the probability thatmore than one Spike could appear in this interval is small enough to be ignored. In this case, theaverage Spike count is equal to the probability of firing a single Spike . That is, the probability of aspike occurring during a given brief time interval is equal to the value of the instantaneous firingrate during that time interval times the length of the interval:Pf1spikeduringtheinterval(t t;t+ t)g=r(t) t:(2)Unlike the neural response function which provides a complete description of the neural re-sponse, the instantaneous firing rate is a highly reduced description. It is constructed by averagingthe neural response function over many repeated trials, to identify the systematic component ofthe response that is common to all trials.

8 Other averages of the neural response function could beconstructed, for example, the response correlation functionh (t) (t0)i. The question is whether ornot it is worth the effort to keep track of anything other than the instantaneous firing ProcessesPoisson processes are important in a variety of problems involving rare, random events in time orspace, , radioactive emissions, traffic accidents, and action Poisson ProcessWe will begin by assuming that the underlying instantaneous firing rateris constant over is called a homogeneous Poisson process. Later we will treat the inhomogeneous case inwhichr(t)varies over time. Imagine that we are given a long interval(0;T)and we place a singlespike in that interval at random. Then we pick a sub-interval(t1;t2)of length t=t2 t1.

9 Theprobability that the Spike occurred during the sub-interval equals t= let s placekspikes in the(0;T)interval and find the probability thatnof them fall in the(t1;t2)sub-interval. The answer is given by the binomial formula:Pfnspikesduring tg=k!(k n)!n!pnqk n;wherep= t=Tandq=1 p. If you have never seen this binomial formula before, look inany undergraduate level probability or statistics book. The binomial formula is what you use tocalculate the probability ofnevents of a certain type out ofktrials, for example, the probability ofgetting 10 sixes out of 100 rolls of a fair we increasekandTkeeping the ratior=k=Tconstant. Sincekis the total numberof spikes andTis the total time,r=k=Tis the mean firing rate, the average number of spikesper second. It can be shown that ask!

10 1, the probability thatnspikes will be in an interval oflength tequals:Pfnspikesduring tg=e r t(r t)nn!:(3)This is the formula for the Poisson probability density function. Given the mean firing rater,the formula tells you the probability of havingnspikes during a time interval of length t. Theformula is only correct when the spikes are completely independent of one another, , that theyare placed randomly throughout the full(0;T)time Spike count for a homogeneous Poisson process, dropping the time-dependence from Eq. 1,is given by:hni=Zt2t1rdt=r t;(4)for any interval of length t=t2 t1. As expected, the average Spike count equals the meanfiring rate times the duration. The variance of the Spike count is a bit harder to derive but it turnsout that the result is the same, , 2n=r t:The ratio of the variance to the mean Spike count is called theFano factor,F= 2nhni=1:(5)The Fano factor characterizes the variability in the Spike count.