Spike Train Statistics Sabri IPM. Review of spike train Extracting information from spike trains Noisy environment: in vitro in vivo measurement.
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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
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]
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