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A model for isolation spiking in sharps

A model for isolation spiking in sharps. Two-part model. Base-spiking or background spiking always present During sharps, an additional sharp-spiking maybe present The two forms of spiking are independent; one doesn’t influence the other. Splitting probabilities. What we measure:

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A model for isolation spiking in sharps

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  1. A model for isolation spiking in sharps

  2. Two-part model • Base-spiking or background spiking always present • During sharps, an additional sharp-spiking maybe present • The two forms of spiking are independent; one doesn’t influence the other

  3. Splitting probabilities • What we measure: • This is roughly the probability of total spiking in sharps • Has two parts: • Probability of base-spiking in sharps • Probability of sharp-spiking in sharps • We need to find the second

  4. Finding the probability • Probability of sharp-spiking in sharps: • Is the probability of 1 or more sharp-spikes in a sharp • Is 1 – (probability of 0 sharp-spikes in a sharp) • Probability of 0 total spikes in a sharp • Is probability of 0 base-spikes in a sharp x probability of 0 sharp-spikes in a sharp (this is the independence assumption) • So, probability of sharp-spiking in sharps: • is • Is • Is

  5. Base spiking model • We just need the probability of 0 base-spikes in a sharp • This is the probability the background contributes nothing to spiking in sharps • Assumption: The overall spiking rate is the background rate of spiking • Let ‘r’ be that rate expressed in units of timestamps, for us roughly 1/2 ms.

  6. Base spiking: binomial distribution • At every time, chances of seeing a background spike is then . • For example, if the average rate were 10 Hz (100 msec average ISI), chances of seeing a spike in a 10 ms window would be 1/10 • Chances of seeing a spike in a timestamp-unit (2 msec) would be 1/50 • Chances of not seeing a spike in a timestamp would be

  7. Base spiking: binomial over a period • If a sharp has ‘t’ timestamps, chances of not seeing any spikes in it • Is chance of no spike in first timestamp x chance of no spike in second timestamp x . . . • So, probability of 0 base-spikes in a sharp of length ‘t’ timestamps

  8. Base spiking: the binomial equation • If we have multiple sharps of lengths t1, t2, so on, with total length of all sharps being ‘t’ • probability of 0 base-spikes in an arbitrary sharp: • This is the weighted average of the individual probabilities, weighted by length of sharp • In math lingo, that is:

  9. Relationship to Poisson • If length of timestamps were reduced from 1/2 ms to 1/4 ms to 1/8 msec, all the way down toward the limit 0 • The base-spiking equation tends toward the Poisson distribution • This is how the Poisson was originally derived • Is the limit case of the binomial distribution as time segment tends to 0 length • No advantage to us in using the Poisson

  10. The full equation • Probability of sharp-spiking in sharps: • where tiis the length of sharp ‘i’ and r is the rate of neuron firing • all time units are in timestamp-units (roughly 1/2 ms for us) • The equation, however, holds for all timeunits where r is less than 1 • As the time unit shortens to 0, the denominator approximates to the Poisson with mean ‘r.’

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