Is the superposition of many random spike trains a Poisson process? Benjamin Lindner Max-Planck-Institut für Physik komplexer Systeme, Dresden Reference: Phys. Rev. E 73, 2291 (26)
Outline Point processes spike train statistics and interval statistics Renewal and nonrenewal processes Poisson process Superposition of many trains - the paradox Resolution of the paradox Conclusions
Stationary point processes Given by an ordered set of random points {t i } t i-3 t i-2 t i-1 t i t i+1 t i+2 t with statistics depending only on time differences (stationarity) We can associate each point with a δ spike spike train x(t) = δ(t t i ) Examples for series of events at random (or random ) instances Emission of an electron (shot noise), occurrences of earthquakes, neural firings (action potentials), breakdown of power supply,
Interval statistics Interspike intervals are: I i = t i+1 t i I i-1 I i I i+1 I i+2 I i+3 t i-1 t i t i+1 t i+2 t i+3 t Can be equivalently described a hierarchy of interval sequences T 3,i {T n,k } T 2,i T 1,i t i-1 t i t i+1 t i+2 t i+3 t
Interval statistics Interspike intervals (ISIs) are: I i = t i+1 t i. Equivalent: n-th order interval T n,k = t i+n t i T 3,i I i-1 I i I i+1 I i+2 I i+3 T 2,i T 1,i t i-1 t i t i+1 t i+2 t i+3 t t i t i+1 t i+2 Serial correlation coefficient measures how strongly the ISIs are correlated t i-1 ρ k = (I i I i )(I i+k I i+k ) (I i I i ) 2 Special class of point processes: renewal processes for which all ISIs are statistically independent PDF of ISI P(I) determines the statistics of the point process completely t i+3 t
Spike train statistics Mean value of spike train = spike rate probability to have a spike in [t,t + dt] is given by r dt For stationary process equivalent to time average TR 1 N(T ) x(t) = lim T T dt δ(t t i ) = lim T T = r
Second-order statistics Spike train statistics II Correlation function: k(τ) = x(t)x(t + τ) x(t) x(t + τ) Can be written as x(t)x(t + τ) = rδ(τ) + rp(τ ) conditional probability of having a spike at t = τ given we p(τ ) observed another one at t = Power spectrum: S( f ) = Z dτe 2πi f τ k(τ) saturates at high frequency at r, i.e. S( f ) = r +
Example Process with reduced probability of short intervals 1 P 1.5 k(τ) 1.5 -.5-1 -2-1 1 2 τ 2 4 6 ISI S(f) 1.5 1.5 2 4 6 f
Relation between ISI and spike-train statistics Given the ISI and n-th order interval densities can we calculate the power spectrum ( f > )? S( f ) = Z Z dtk(t)e 2πi ft = r + 2rR Z = r + 2rR = r [ 1 + n=1 dt 2πi ft P n (t)e n=1 ] P n ( f ) + P n ( f ) 2πi ft dt p(t )e For a renewal process: P n ( f ) = P 1 n( f ) S( f ) = [ ( ) ( )] 1 r 1 + 1 P 1 ( f ) 1 1 + 1 P 1 ( f ) 1 = r 1 P 1 ( f ) 2 1 P 1 ( f ) 2
The simplest process Poissonian shot noise Completely random occurrence of points No correlations between spikes flat power spectrum k(τ) = rδ(τ), S( f ) = r Exponential ISI density, ISIs are uncorrelated (special renewal process) P(I) = r exp[ ri], ρ k =,k > PDF for spike count (number of events in [,T ]) P T (N) = (rt )N exp[ rt ] N!
Three ways of generating a Poisson process Distribute N points independently and uniformly on the interval [, T ]. For a time window [,T ] with T T the resulting point process will be Poissonian. Go through the interval [,T ] by steps of t 1/r. In each step draw an independent random number ξ i uniformly distributed in [,1]. If ξ i < r t, there is an event, otherwise not. Draw a sequence of independent random numbers according to the exponential PDF r exp[ rt]. The sequence of spike times is then given by t i = i j=1 T j.
Is the superposition of many trains Poissonian? Given a large number of independent non-poissonian spike trains: Is their superposition a Poisson process? Is there a central limit theorem for point processes? Important, for instance, for the problem of estimating the effective input of a neuron connected to 1 4 other neurons
Apparently it is... If the total rate of X(t) = N j=1 x j (t) which is r = N j=1 r j remains constant in the limit N, X(t) approaches a Poisson process in this limit [additional assumptions on x j (t) exclude some pathological cases] More recent claims Classical proof by Khinchine (196) Similarly to the central limit theorem [for random variables]..., the sum of independent renewal point processes tends to a Poisson process... (Shimokawa et al., PRE 1999) Since the sum of independent renewal processes tends to a Poisson process, the superposition of a large number of output spike trains can be approximated by a... Poisson process... (Hohn & Burkitt, PRE, 21)
Numerical results seem to support it... Numerics shows that for N we get an exponential ISI and a flat power spectrum, i.e. a Poisson process. Power spectrum of summed spike train (Hohn & Burkitt, PRE, 21) lower curve: spectrum of single spike train, because of refractory period it shows a dip at low frequencies upper curve: spectrum of the superposition of 1 independent spike trains, apparently the dip starts to vanish and the spectrum becomes flat also at low frequencies.
My test model Single spike train is a renewal but non-poissonian process with ISI density and power spectrum p α (T ) = 4r 2 T exp[ 2rT ], [ 2r 2 ] S α = r 1 4r 2 + (π f ) 2 Consider X(t) = 1 N N x n (t) n=1 (prefactor to keep the mean value of the spike train constant, not of any relevance for the problem) Is X(t) a Poisson spike train?
ISI statistics why a Poissonian statistics can be naively concluded from it Spike trains PDF of ISI Correlation coefficient 1 4 1 N=1 N=1.1 N=1.5 1 2 T i T i+1 -.1 1 4 1 N=2 N=2.1 N=2.5 1 2 -.1 1 4 1 N=1 N=1.1 N=1.5 1 2 -.1 1 4 1 N=1 N=1.1 N=1.5 1 2 -.1 1 2 2 4 6 8 1 time k 2 4 6 N T i For large N ISI density approaches a simple exponential, ISI correlations for any finite lag vanish X(t) approaches a renewal process with exponential ISI density, i.e. a Poisson process
Spectral statistics why X(t) is not a Poissonian spike train Correlation function of summed spike train K X (τ) = X(t)X(t + τ) X(t) X(t + τ) = 1 N 2 x n (t)x l (t + τ) x n (t) x l (t + τ) n,l = 1 N 2 [ x n (t)x n (t + τ) x n (t) x n (t + τ) ] + 1 n N 2 [ x n (t)x l (t + τ) x n (t) x l (t + τ) ] n l Hence K X (τ) = 1 N K x(τ) S X ( f ) = 1 N S x( f ) N S X (f) 1.5 1.5 N=1 N=2 2 4 6 f N=1 N=1 2 4 6 f Similar recent finding by Cateau & Reyes Phys. Rev. Lett. 96, 5811 (26)
Result for the power spectrum by Hohn and Burkitt? Reminder: H&B considered output of 1 leaky integrate-and-fire neurons driven by white noise and used X(t) = x(t) [not X(t) = 1 N x(t)] Hence S X ( f ) = NS x ( f ) power spectra (db) 8 6 4 2 S 1 (f) S 1 (f) S 1 (f)+3-2.5 1 1.5 2 2.5 f Shifted curve is in good agreement with the spectrum of the single spike train. Result by Hohn and Burkitt is a numerical artifact!
The paradox On the one hand... ISI statistics tells us that we deal for large N with a renewal process with exponential density. A renewal process is completely determined by its density and thus if its density agrees with that of a Poisson process, then it is a Poisson process. On the other hand... The power spectrum of X(t) for arbitrary N is exactly proportional to that of the single spike train, it is not flat for a non-poissonian x(t) and thus X(t) is not a Poisson process.???
Resolution of the paradox X(t) is not a renewal process even in the limit N but a process with infinitesimally small correlations between the ISIs for infinitely many lags Spectrum at zero frequency, where S α () = r/2 while for a Poisson shot noise we would have S poi = r. At zero frequency, we have S() = r 3 (T i T i ) 2 [ 1 + 2 k=1 ρ k ]. (for exp. ISI statistics (T i T i ) 2 = r 2 ) Sum over correlation coefficients makes a finite impact ρ k 1/4 for N k=1 m -.5 -.1 N=1 Σρ N=2 k k=1 -.15 -.2 -.25 N=1 N=1 2 4 6 8 1 m
Sticking to Khinchine s exact condition Diluting the single spike train when adding more and more trains such that the total rate is conserved ˆX = x n (Nt) Then S X (f) ^ 1.5 1.5 N=1 N=2 2 4 f N=1 N=1 2 4 f which is plainly so because the spectrum of the single spike train tends to a flat spectrum at fixed frequency (interesting spectral features are shifted to lower and lower frequencies) However, not relevant for the neural (and other) applications!
Khinchine s exact condition are not met for neurons! Input rate of a single presynaptic neuron output rate of the postsynaptic neuron Output rate is a functional of the superposed input spike trains
Feed-forward network - transmission of transient stimuli FIG. 2. Simulated and predicted Cateau behavior & Reyes of feedforward Phys. Rev. net- Lett. 96, 5811 (26) Left: Simulation of the time-dependent firing rate for different layers, synchrony among neurons increases while progressing through the different layers Right: Theoretical predictions of firing rates grey: using the Poisson approximation, the rate flattens out (no synchrony), black:: taking the spectral features of the superposed spike train into account yields increasing synchrony as in the simulations
Conclusions Be careful and cautious with limit theorems! Check all the models that are based on the Poisson assumption (diffusion approximation)