The homogeneous Poisson process

Transcription

1 The homogeneous Poisson process during very short time interval Δt there is a fixed probability of an event (spike) occurring independent of what happened previously if r is the rate of the Poisson process, then the probability of finding a spike in a short interval Δt is given by r Δt. ( ) The probability of seeing exactly n spikes in a (long) interval T is given by the Poisson distribution: = = (r ) ( r )

2 Derivation Divide time T into bins Δt=T/M. The probability of a spike occurring in one specific bin is rδt P T [n] is product of three factors: 1. probability of generating n spikes within a specified set of the M bins. 2. probability of not generating spikes in the remaining M n bins 3. combinatorial factor equal to number of ways of putting n spikes into M bins Thus, the probability of seeing exactly n spikes in T is = (binomial distribution) ( ) (r ) ( r )

3 Derivation (cont.) For Δt->0, M=T/Δt grows without bound and thus: Further: = ( ) (r ) ( r ) M! (M n)! = M(M 1)(M 2) (M n + 1) M n = T n t n ( ) ln(1 r t) M n =(M n)ln(1 r t) Mr t = Tr and thus (1 r t) M n = e Tr = Therefore = (r ) ( r )

4 Gaussian fit ( ) = = (r ) Properties: mean: E[n] = rt variance: E[ (n - E[n]) 2 ] = rt Fano factor: variance/mean = 1 ( r ) good approximation by Gaussian for large rt (see fig. B)

5 Interspike interval distribution of homogeneous Poisson process Suppose a spike occurred at time t i Probability of generating next spike somewhere in interval is equal to the probability that no spike is fired for a time τ, times the probability, rδt, of generating spike within the following Δt. + Thus + τ + < + τ + τ + < τ + = r ( rτ) The probability density of interspike intervals is, by definition, this probability with the factor Δt removed:

6 Interspike interval distribution of homogeneous Poisson process is given by exponential distribution (short interspike intervals frequent, long ones rare): Mean: Variance: Coefficient of variation: Note: Fano factor and C V of 1 are necessary but not sufficient for a Poisson process!

7 Comparison of Poisson model with data: mean and variance fit data to model: Poisson: A=B=1 A: 94 cells macaque MT, 256ms, various conditions: Fano factor (slope) is about one B, C: parameters A and B as a function of the duration of the counting window

8 Comparison with data: inter-spike interval distribution from monkey MT neuron for moving random-dot image Poisson: distribution should be exponential Experimental: absolute and relative refractory period produce shortage of small inter-spike intervals (left) Improved model (right): use refractory period of variable duration (Gaussian with mean 5 ms and standard deviation 2ms) after which Poisson model is used again.

9 The Inhomogeneous Poisson process when firing rate depends on time situation is more complicated: probability of each spike depends on current firing rate r(t) Poisson Spike Generator (to create artificial spike trains): first calculate estimate of firing rate r est (t) for every time t divide time into short intervals Δt for every interval generate spike with probability r est (t) Δt by simply drawing a uniform number from [0, 1] and checking if it is smaller than r est (t) Δt. In that case generate a spike. r est (t) may be derived from knowledge of stimulus and fitted response tuning function Note: to capture refractory effects, can set rate to zero after spike and then let it exponentially return to predicted value

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11 constant current constant current moving grating Notes: intracellular recordings from cat V1 neurons simple Poisson model provides reasonable fit in many but not all situations does not provide mechanistic explanation of where variability comes from. In fact, spike generation in neurons seems to be a quite reliable process (left) in vivo, however, things tend to look more irregular some neurons tend to fire bursts of spikes

12 Spike-Train Autocorrelation Function interspike interval distribution relates times of successive spikes. Let s generalize to relation between any spikes autocorrelation = of the neural response function with its average over time and trials subtracted out: ρρ (τ) = (ρ( ) )(ρ( + τ) ) symmetry: for a homogeneous Poisson process (recall: no dependencies between events): due to mean subtraction this really is a covariance!

13 Cross-correlation function (generally not symmetric): Figure: auto-correlation (A) and cross-correlation (B) functions indicating synchronous oscillatory activity at 40Hz (gamma band) across both brain hemispheres; Neurons from cat V1, left and right hemisphere.

14 1.5 The Neural Code

15 The Neural Code Main ideas: Independent-Spike code Independent-Neuron code Correlation codes synchrony and oscillations temporal code

16 Independent Spike Code: consider spike generation due to a Poisson process: in this case the time-dependent firing rate r(t) contains all the information about the stimulus that can be extracted from the spike train. Relative timing of spikes contains no additional information about stimulus. Correlation Code: individual spikes do not encode independently of each other. Correlations between spikes may carry additional information about the stimulus. The amount of this information should be significant to warrant using the term correlation code. Example: information could be encoded in the duration of inter-spike intervals Although some extra information may be in correlations between spikes, independent spike-coding seems reasonable approximation

17 Independent Neuron Code: assumes that neurons act independently. This does not mean that spike trains from different neurons are not combined into an ensemble code, it just means that the code can be decoded without taking correlations between neurons into account. Synchronous firing of two or more neurons is one mechanism for conveying extra information compared to an independent neuron code. Synchronous firing and timelocked oscillations are frequent, but the presence of synchronicity by itself does not mean that these correlations carry additional information.

18 Hippocampal Place Cells: One example where additional information seems to be carried by correlations between firing patterns within a population. The phase of firing within a theta cycle correlates with the position of the animal

19 Temporal Codes: How precisely must we measure spike times or ratefluctuations in order to see all the information contained in the spike train about the stimulus? If high precision required we might say: temporal code But: is structure at high temporal frequency solely resulting from the dynamics of the stimulus? Maybe we should require information carried by temporal fine structure that is clearly faster than any variations in the stimulus.

20 time-dependent firing of an MT neuron in response to three different time-varying stimuli. rate code? temporal code?

21 1.6 Chapter Summary

22 Key Concepts neuron, synapse, neurotransmitter, spike, spike train different notions of firing rate tuning curve spike-triggered average white noise stimulus homogeneous Poisson process interspike interval distribution spike train autocorrelation inhomogeneous Poisson process Poisson spike generator neural code