Spike Train Statistics

SabriIPM

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 Methodsrecording

stimulus

repeated trials

stimulus onset stimulus onset

Firing rate estimation methods:• PSTH• Kernel density function

Information is in the difference of firing rates over time

Parametric Methodsrecording

stimulus

repeated trials

stimulus onset stimulus onset

Fitting P distribution with parameter set: …,

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:

[]P

[]p

: probability of an event (a single spike)

: probability density function

nnn ttttptttP ,,,,,, 2121

Spike time: ttt ii ,

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

ttiti-1

Firing rate: The probability of firing a single spike in

a small interval around ti

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

trrepeated trials

Homogeneous Distributions: firing rate is considered constant over timeInhomogeneous 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:

rtr

0 T

….

t1 ti tn

n

Tnn T

tnPntttPtttP

!,,,,,, ''2

'121

rTT e

n

rTnP

! : 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:

nn2

rTnn 2

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:ti ti+1

Interspike interval

rrep rii tertttP

1

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

vC

1vC a necessary but not sufficient condition to identify Poisson spike train

2vC

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 t0 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:

ttrttttP ii tttHttttP ii 0

dHHp

0exp

ReR

Rp

1

1vC

2

2

2

logexp

2

1

p 1exp 2 vC 2exp 2 R

R

RC

R

Cp vv

22

3

1 1

2exp

2

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

nnnnp ,,,| 21

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]