Transcript Spike Train Statistics

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
Firing rate estimation methods:
• PSTH
• Kernel density function
Information is in the difference
of firing rates over time
stimulus onset
stimulus onset
Parametric Methods
recording
stimulus
repeated trials
stimulus onset
stimulus onset
Fitting P distribution with parameter set: 𝜃1 , 𝜃2 , …,𝜃𝑚
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:
P[] : probability of an event (a single spike)
p[] : probability density function
 Probability corresponding to every sequence of spikes that
can be evoked by the stimulus:
Joint probability of n events at
specified times
Pt1, t2 ,, tn   pt1, t2 ,, tn  t 
n
Spike time:
ti , ti  t 
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
ti-1 ti
t
Firing rate: r t 
 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
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: r t   r the probability of firing is constant
during period T
 Each sequence probability:
….
0 t1
ti

tn

T
 t 
' '
'
Pt1 , t2 ,, tn   P t1 , t2 ,, tn  n! PT n  
T 
rT  e rT
PT n 
: Probability of n events in [0 T]
n!
n
[Dayan and Abbot, 2001]
rT=10
Fano Factor
 Distribution Fitting validation
 The ratio of variance and mean of the spike count
2
 For homogenous Poisson model:  n  n  rT
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]
 n2 n
Interspike Interval (ISI) distribution
 Distribution Fitting validation
 The probability density of time intervals between adjacent spikes
Interspike interval

ti
ti+1
 for homogeneous Poisson model: P  ti 1  ti    t   rter  p   re r
MT neuron
[Dayan and Abbot, 2001]
Poisson model with a stochastic refractory period
Coefficient of variation
 Distribution Fitting validation
 In ISI distribution: Cv   
 For homogenous Poisson: Cv  1
a necessary but not sufficient condition to
identify Poisson spike train
 For any renewal process, the Fano Factor over long time intervals
approaches to value Cv2
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: Pti  t  ti  t   r t   t 
 For renewal processes: Pti  t  ti  t   H t  t0   t 
 in which t0 is the time of last spike
 And H is hazard function


 By these definitions ISI distribution is: p   H   exp   0 H   d  


 Commonly used renewal processes:
 Gamma process: (often used non Poisson process)

 R  1  R
p    R
e
 
Cv  1 
 Log-Normal process:
 log   2 
1

p  
exp 
2

2
2 


 Inverse Gaussian process:
 Cv2 R  12 
Cv1

p  
exp 

3
2
R

2 R




R  exp     2 2
 
Cv  exp  2  1
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
spike train from rat CA1
hippocampal pyramidal neurons
recorded while the animal
executed a behavioral task
Inhomogeneous Gamma
Inhomogeneous inverse Gaussian
[Riccardo et al., 2001, J. Neurosci. Methods]
Inhomogeneous Poisson
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
p n |  n1 , n2 ,, n 
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]