<p>First of all you should notice the assumption that spikes are independent and that every sequence of spikes over an interval has the same probability. That's why you can express <span class="math-container">$P[t_{1}, t_{2}, ..., t_{n}]$</span> in terms of <span class="math-container">$P_{T}[n]$</span>. 
Second, note that <span class="math-container">$p[t_{1}, t_{2}, ..., t_{n}]$</span> is a probability density. So you can express the probability as the integral of <span class="math-container">$p[t_{1}, t_{2}, ..., t_{n}] dt_{1}dt_{2}...dt_{n}$</span></p>

<p><span class="math-container">$\Delta t / T$</span> is the probability that one spike occurs in that time interval. If <span class="math-container">$\Delta t$</span> is small enough, only one spike is within that interval, so just let <span class="math-container">$\Delta t \rightarrow dt_{i}$</span>. Then  <span class="math-container">$p[t_{1}, t_{2}, ..., t_{n}] dt_{1}dt{2}...dt{n} = n!P_{T}[n]dt_{1}dt_{2}...dt_{n}(\frac{1}{T})^{n}$</span></p>

<p>You should see this as a joint probability.</p>

CC BY-SA 4.0
2019-08-23T08:22:37.627Z
2019-08-23T08:22:37.627Z
julix
<p>In Theoretical Neuroscience by Dayan and Abbott page 25, </p>

<blockquote>
  <p>the probability <span class="math-container">$P[t_1, t_2, ..., t_n]$</span> that a sequence of n spikes occurs with spike <span class="math-container">$i$</span> falling between times <span class="math-container">$t_i$</span> and <span class="math-container">$t_i+\Delta t$</span> for <span class="math-container">$i=1,2,...,n$</span> is given in terms of this by the relation <span class="math-container">$P[t_1, t_2, ..., t_n]=p[t_1,...,t_n](\Delta t)^n$</span>... the probability <span class="math-container">$P[t_1, t_2, ..., t_n]$</span> can be expressed in terms of another probability function <span class="math-container">$P_T[n]$</span>, which is the probability that an arbirary sequence of exactly n spikes occurs within a trial of duration T. Assuming that the spike times are ordered so that <span class="math-container">$0 \le t_1 \le t_2 \le ... \le t_n \le T$</span>, the relationship is 
  <span class="math-container">$$P[t_1, t_2, ..., t_n] = n!P_T[n]\big({\Delta t \over T}\big)^n$$</span></p>
</blockquote>

<p>I get that <span class="math-container">$P_T[n]$</span> is the probability function of <span class="math-container">$n$</span> spikes occurring(which later is proved to be a Poisson distribution).  I'm guessing <span class="math-container">$n!$</span> is different ways of re-arranging the <span class="math-container">$n$</span> spikes. What exactly is <span class="math-container">$\big({\Delta t \over T}\big)^n$</span>? </p>

<theoretical-neuroscience><computational-modeling>
Homogeneous Poisson Process of spike train
MoneyBall
# Homogeneous Poisson Process of spike train

(Asked by: username_0 on 2019-05-02)

In Theoretical Neuroscience by Dayan and Abbott page 25,

> the probability $P[t\_1, t\_2, ..., t\_n]$ that a sequence of n spikes occurs with spike $i$ falling between times $t\_i$ and $t\_i+\Delta t$ for $i=1,2,...,n$ is given in terms of this by the relation $P[t\_1, t\_2, ..., t\_n]=p[t\_1,...,t\_n](\Delta t)^n$... the probability $P[t\_1, t\_2, ..., t\_n]$ can be expressed in terms of another probability function $P\_T[n]$, which is the probability that an arbirary sequence of exactly n spikes occurs within a trial of duration T. Assuming that the spike times are ordered so that $0 \le t\_1 \le t\_2 \le ... \le t\_n \le T$, the relationship is
> $$P[t\_1, t\_2, ..., t\_n] = n!P\_T[n]\big({\Delta t \over T}\big)^n$$

I get that $P\_T[n]$ is the probability function of $n$ spikes occurring(which later is proved to be a Poisson distribution). I'm guessing $n!$ is different ways of re-arranging the $n$ spikes. What exactly is $\big({\Delta t \over T}\big)^n$?

**Tags:** theoretical-neuroscience, computational-modeling


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## 1 answers

### Answer by username_1 on 2019-08-23. Score: 2

First of all you should notice the assumption that spikes are independent and that every sequence of spikes over an interval has the same probability. That's why you can express $P[t\_{1}, t\_{2}, ..., t\_{n}]$ in terms of $P\_{T}[n]$.
Second, note that $p[t\_{1}, t\_{2}, ..., t\_{n}]$ is a probability density. So you can express the probability as the integral of $p[t\_{1}, t\_{2}, ..., t\_{n}] dt\_{1}dt\_{2}...dt\_{n}$

$\Delta t / T$ is the probability that one spike occurs in that time interval. If $\Delta t$ is small enough, only one spike is within that interval, so just let $\Delta t \rightarrow dt\_{i}$. Then $p[t\_{1}, t\_{2}, ..., t\_{n}] dt\_{1}dt{2}...dt{n} = n!P\_{T}[n]dt\_{1}dt\_{2}...dt\_{n}(\frac{1}{T})^{n}$

You should see this as a joint probability.


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