Spike Train Statistics Sabri IPM

Review of spike train  Extracting information from spike trains  Noisy environment:  in vitro  in vivo  measurement  unknown inputs and states  what kind of code:  rate: rate coding (bunch of spikes)  spike time: temporal coding (individual spikes) [Dayan and Abbot, 2001]

Non-parametric Methods recording stimulus repeated trials stimulus onset Firing rate estimation methods: PSTH Kernel density function Information is in the difference of firing rates over time

Parametric Methods recording stimulus repeated trials stimulus onset Two sets of different values for two raster plots Parameter estimation methods: ML – Maximum likelihood MAP – Maximum a posterior EM – Expectation Maximization

Models based on distributions: definitions & symbols  Fitting distributions to spike trains:  Probability corresponding to every sequence of spikes that can be evoked by the stimulus: : probability of an event (a single spike) : probability density function Spike time: Joint probability of n events at specified times

Discrete random processes  Point Processes:  The probability of an event could depend of the entire history of proceeding events  Renewal Processes  The dependence extends only to the immediately preceding event  Poisson Processes  If there is no dependence at all on preceding events t titi t i-1

Firing rate:  The probability of firing a single spike in a small interval around t i  Is not generally sufficient information to predict the probability of spike sequence  If the probability of generating a spike is independent of the presence or timing of other spikes, the firing rate is all we need to compute the probabilities for all possible spike sequences repeated trials Homogeneous Distributions: firing rate is considered constant over time Inhomogeneous Distributions: firing rate is considered to be time dependent

Homogenous Poisson Process  Poisson: each event is independent of others  Homogenous: the probability of firing is constant during period T  Each sequence probability: 0T …. t1t1 titi tntn : Probability of n events in [0 T] rT=10 [Dayan and Abbot, 2001]

Fano Factor  Distribution Fitting validation  The ratio of variance and mean of the spike count  For homogenous Poisson model: MT neurons in alert macaque monkey responding to moving visual images: (spike counts for 256 ms counting period, 94 cells recorded under a variety of stimulus conditions) [Dayan and Abbot, 2001]

Interspike Interval (ISI) distribution  Distribution Fitting validation  The probability density of time intervals between adjacent spikes  for homogeneous Poisson model: titi t i+1 Interspike interval MT neuron Poisson model with a stochastic refractory period [Dayan and Abbot, 2001]

Coefficient of variation  Distribution Fitting validation  In ISI distribution:  For homogenous Poisson:  For any renewal process, the Fano Factor over long time intervals approaches to value a necessary but not sufficient condition to identify Poisson spike train Coefficient of variation for V1 and MT neurons compared to Poisson model with a refractory period: [Dayan and Abbot, 2001]

Renewal Processes  For Poisson processes:  For renewal processes:  in which t 0 is the time of last spike  And H is hazard function  By these definitions ISI distribution is:  Commonly used renewal processes:  Gamma process: (often used non Poisson process)  Log-Normal process:  Inverse Gaussian process:

ISI distributions of renewal processes [van Vreeswijk, 2010]

Gamma distribution fitting spiking activity from a single mushroom body alpha-lobe extrinsic neuron of the honeybee in response to N=66 repeated stimulations with the same odor [Meier et al., Neural Networks, 2008]

Renewal processes fitting [Riccardo et al., 2001, J. Neurosci. Methods] spike train from rat CA1 hippocampal pyramidal neurons recorded while the animal executed a behavioral task Inhomogeneous Poisson Inhomogeneous Gamma Inhomogeneous inverse Gaussian

Spike train models with memory  Biophysical features which might be important  Bursting: a short ISI is more probable after a short ISI  Adaptation: a long ISI is more probable after a short ISI  Some examples:  Hidden Markov Processes:  The neuron can be in one of N states  States have different distributions and different probability for next state  Processes with memory for N last ISIs:  Processes with adaptation  Doubly stochastic processes

Take Home messages  A class of parametric interpretation of neural data is fitting point processes  Point processes are categorized based on the dependence of memory:  Poisson processes: without memory  Renewal processes: dependence on last event (spike here)  Can show refractory period effect  Point processes: dependence more on history  Can show bursting & adaptation  Parameters to consider  Fano Factor  Coefficient of variation  Interspike interval distribution

Spike train autocorrelation  Distribution of times between any two spikes  Detecting patterns in spike trains (like oscillations) Autocorrelation and cross-correlation in cat’s primary visual cortex: Cross-correlation: a peak at zero: synchronous a peak at non zero: phase locked [Dayan and Abbot, 2001]

Neural Code  In one neuron:  Independent spike code: rate is enough (e.g. Poisson process)  Correlation code: information is also correlation of two spike times (not more than 10% of information in rate codes, Abbot 2001)  In population:  Individual neuron  Correlation between individual neurons adds more information  Synchrony  Rhythmic oscillations (e.g. place cells) [Dayan and Abbot, 2001]