Biomedical Engineering Reference
In-Depth Information
8.3.1.2
Transition Probabilities
The probability that a neuron emits a spike at some time n depends on the history
of the neural network. However, it is impossible to know explicitly its form in the
general case since it depends on the past evolution of all variables determining the
neural network state. A possible simplification is to consider that this probability
depends only on the spikes emitted in the past by the network. In this way, we
are seeking a family of transition probabilities of the form P ω ( n ) ω n− 1
n−D ,the
probability that the firing pattern ω ( n ) occurs at time n , given a past spiking
sequence ω n− 1
n−D .Here D is the memory depth of the probability, i.e., how far in
the past does the transition probability depend on the past spike sequence. We use
here the convention that P ω ( n ) ω n− 1
n−D = P [ ω ( n )] if D =0(memory-less
case).
Transition probabilities depend on the neural network characteristics such as neu-
rons conductances, synaptic responses or external currents. They give information
on the dynamics that takes place in the observed neural networks. Especially, they
have a causal structure where the probability of an event depends on the past. This
reflects underlying biophysical mechanisms in the neural networks which are also
causal. The explicit computation of transition probabilities can be done in some
model-examples (Sect. 8.4.4 ). From them, one is able to characterize statistical
properties of rasters generated by the network, as we now develop.
8.3.1.3
Markov Chains
Transition probabilities with a finite memory depth D define a “Markov chain”, i.e.,
a random process where the probability to be in some state at time n (here a spiking
pattern ω ( n )) depends only upon a finite past (here on ω ( n
D )).
Markov chains have the following property. Assume that we know the probability
of occurrence of the block ω m + D− 1
m
1) ,...,ω ( n
,
P ω m + D− 1
m
= P [ ω ( m + D
1) ( m + D
2) ,...,ω ( m )] .
(8.1)
Then, by definition, the probability of the block ω m + D
m
is:
P ω m + D
m
= P [ ω ( m + D ) ( m + D
1) ,...,ω ( m )]
= P [ ω ( m + D ) |
ω ( m + D
1) ,...,ω ( m )]
×
P [ ω ( m + D
1) ,...,ω ( m )] .
Thus:
P ω m + D
m
= P ω ( m + D ) ω m + D− 1
m
P ω m + D− 1
m
,
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