Sabri
IPM

 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]

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

recording stimulus repeated trials stimulus onset stimulus onset
Fitting P distribution with parameter set: 𝜃
1
, 𝜃
2
, …, 𝜃 𝑚
Parameter estimation methods:
• ML – Maximum likelihood
• MAP – Maximum a posterior
• EM – Expectation Maximization
Two sets of different values for two raster plots

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
 t
1
, t
2
,  , t n
  t
1
, t
2
,  ,
Spike time:
 t i
, t n t i
   n
  t


 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 i-1 t i t

r
 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

 Poisson: each event is independent of others
 Homogenous: r
 
 r the probability of firing is constant during period T
 Each sequence probability:
….
0 t
1 t i t n
T
P
 t
1
, t
2
,  , t n


P
 t
1
'
, t
2
'
,  , t n
'

 n !
P
T
 
 t
T n
P
T
 n !
e
 rT
: Probability of n events in [0 T]
[Dayan and Abbot, 2001] rT=10

 Distribution Fitting validation
 The ratio of variance and mean of the spike count
 For homogenous Poisson model:

2 n
 n
 rT
 n
2 n
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]

 Distribution Fitting validation
 The probability density of time intervals between adjacent spikes
Interspike interval
 t i t i+1
 for homogeneous Poisson model: P

  t i

1
 t i
    t

 r
 te
 r
  p
 
 re
 r

MT neuron
[Dayan and Abbot, 2001]
Poisson model with a stochastic refractory period

 Distribution Fitting validation
 In ISI distribution:
C v
 


 For homogenous Poisson: C v

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 C v
2
Coefficient of variation for V1 and
MT neurons compared to Poisson model with a refractory period:
[Dayan and Abbot, 2001]

 For Poisson processes: P

 For renewal processes: P t i
 t i

 t t

 t i t i


 t
 t



 r
   
H
 t

 t
0
  
 in which t
0 is the time of last spike
 And H is hazard function
 By these definitions ISI distribution is: p
 

H
  exp
 Commonly used renewal processes:
 Gamma process: (often used non Poisson process) p
 
 
R
 
R
 

 
 
1 e
 
R

C v

1

 Log-Normal process: p
 

2
1
  exp




 log

2


2
 
2



R
 exp

   
2
2

  
0
H
  d
  


C v
 exp

1
 Inverse Gaussian process: p
 

2
C
 v

1
R

3 exp



C v

2
2

R
 
1

2
R




[van Vreeswijk, 2010]

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]

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

 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: p

 n
|
 n

1
,
 n

2
,  ,
 n

 Processes with adaptation
 Doubly stochastic processes

 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

 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]

 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]