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Neural Codes

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Neuronal Codes – Action potentials as the elementary units

voltage clamp from a brain cell of a fly

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Neuronal Codes – Action potentials as the elementary units

voltage clamp from a brain cell of a fly

after band pass filtering

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Neuronal Codes – Action potentials as the elementary units

voltage clamp from a brain cell of a fly

after band pass filtering

generated electronicallyby a threshold discriminatorcircuit

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Neuronal Codes – Probabilistic response and Bayes’ rulestimulus

)(|}{ tstP i

stimulusspiketrains

conditional probability: Given a stimulus, how likely is it to observe a certain spike train?

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Neuronal Codes – Probabilistic response and Bayes’ rule

conditionalprobability

)(tsPensembles of signals

natural situation:

)(},{ tstP ijoint probability:

experimental situation:

• we choose s(t)

)()(|}{)(},{ tsPtstPtstP ii prior

distributionjoint

probability

Asking (left side): How frequently do we observe stimulus and spike train TOGETHER? Answer (right side): As often as this spike train probably follows the stimulus presentation (1st term) times the probability of presenting the stimulus out of many stimuli which we use (right term).

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Conditional probability is the probability of some event A, given theoccurrence of some other event B. Conditional probability is writtenP(A|B), and is read "the probability of A, given B".

Joint probability is the probability of two events in conjunction. That is, itis the probability of both events together. The joint probability of A and Bis written P(A,B).

Consider the simple scenario of rolling two fair six-sided dice, labelled die 1 and die 2. Define the following three events:

A: Dice 1 lands on 3.B: Dice 2 lands on 1.C: The dice sum to 8.

The prior probability of each event describes how likely the outcome is before thedice are rolled, without any knowledge of the roll's outcome. For example, die 1 isequally likely to fall on each of its 6 sides, so P(A) = 1 / 6. Similarly P(B) = 1 / 6.

Likewise, of the 6 × 6 = 36 possible ways that two dice can land, and only 5 of them result in a sum of 8 (namely 2 and 6, 3 and 5, 4 and 4, 5 and 3, and 6 and 2), so P(C) = 5 / 36.

)(tsPPrior

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A: Dice 1 lands on 3.B: Dice 2 lands on 1.C: The dice sum to 8.

Some of these events can both occur at the same time; for example events A and C can happen at the same time, in the case where dice 1 lands on 3 and dice 2 lands on 5. This is the only one of the 36 outcomes where both A and C occur, so its probability is 1/36. The probability of both A and C occurring is called the jointprobability of A and C and is written P(A,C)=1/36. On the other hand, if dice 2 lands on 1, the dice cannot sum to 8, so P(B,C)=0.

Now suppose we roll the dice and cover up dice 2, so we can only see dice 1, and observe that dice 1 landed on 3. Given this partial information, the probability thatthe dice sum to 8 is no longer 5/36; instead it is 1/6, since dice 2 must land on 5 to achieve this result. This is called the conditional probability, because it's theprobability of C under the condition that is A is observed, and is written P(C|A), which is read "the probability of C given A.

On the other hand, if we roll the dice and cover up dice 2, and observe dice 1, thishas no impact on the probability of event B, which only depends on dice 2. We sayevents A and B are statistically independent or just independent and in this case: P(B|A)=P(B).

)(},{ tstP iJoint

Conditional P[{ti}|s(t)]

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Neuronal Codes – Probabilistic response and Bayes’ rule

• But: the brain “sees” only {ti}• and must “say” something about s(t)

• But: there is no unique stimulus in correspondence with a particular spike train• thus, some stimuli are more likely than others given a particular spike train

experimental situation: )()(|}{)(},{ tsPtstPtstP ii

response-conditional ensemble

}{}{|)()(},{ iii tPttsPtstP

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Neuronal Codes – Probabilistic response and Bayes’ rule

)()(|}{)(},{ tsPtstPtstP ii

}{}{|)()(},{ iii tPttsPtstP

)()(|}{}{}{|)( tsPtstPtPttsP iii

}{

)()(|}{}{|)(i

ii tPtsPtstPttsP

Bayes’ rule:

what we see:

what ourbrain “sees”:

What is the difference: Fundamentally WE know the prior P(s) as WE choose the stimuli. The Brain knows the Prior P(t) as the Brain generates the spike trains.

What would the animal (the percept) like to know? It would like to know: Given a spike train what is the most likely stimulus behind it? This is P(s|t).

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Neuronal Codes – Probabilistic response and Bayes’ rule

motion sensitive neuron H1 in the fly’s brain:

average angular velocityof motion across the View Field

in a 200ms window

spike count

determined by the experimenter

property of theneuron

)()(, vPnPvnP Correlation, Not (!) independent

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Neuronal Codes – Probabilistic response and Bayes’ rule

}{|)( ittsP

spikes

determine the probability of astimulus from given spike train

stimuli

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Neuronal Codes – Probabilistic response and Bayes’ rule

}{|)( ittsPdetermine the probability of astimulus from given spike train

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Neuronal Codes – Probabilistic response and Bayes’ rule

)(|}{ tstP i

determine probability ofa spike trainfrom a given stimulus

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Neuronal Codes – Probabilistic response and Bayes’ rule

)(|}{ tstP i

)(tr

determine probability ofa spike trainfrom a given stimulus

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Neuronal Codes – Probabilistic response and Bayes’ rule

)(trHow do we measure this time dependent firing rate?

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Neuronal Codes – Probabilistic response and Bayes’ rule

Nice probabilistic stuff, but

SO, WHAT?

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Neuronal Codes – Probabilistic response and Bayes’ rule

SO, WHAT?

We can characterize the neuronal code in two ways:

translating stimuli into spikes translating spikes into stimuli

}{|)( ittsP )(|}{ tstP i

}{

)()(|}{}{|)(i

ii tPtsPtstPttsP Bayes’ rule:

(traditional approach)

-> If we can give a complete listing of either set of rules, than we can solve any translation problem

• thus, we can switch between these two points of view

(how the brain “sees” it)

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Neuronal Codes – Probabilistic response and Bayes’ rule

We can switch between these two points of view.

And why is that important?

These two points of view may differ in their complexity!

Traditionally you would record this:

Spike count n dependent on the Stimulus (velocity v).

This is a difficult „curved“ function and requires a complex model to explain, does‘nt it??

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Lets measure this in a better (more complete way):

You choose P(v) and for some reason you like some stimuli better thanothers , which makes this peaked.

Do this for all stimulusand don‘t forget to normalize all thisto 1 before plotting(P(n,v).

Then you record the responses (spike count n) for these stimuli. For example the red stimulus gives you after many repetitions the red responsecurve.

Do this for all stimulusand don‘t forget to normalize all thisto 1 before plotting(P(n,v).

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Lets measure this in a better (more complete way):

You choose P(v) and for some reason you like some stimuli better thanothers , which makes this peaked.

Do this for all stimuliand don‘t forget to normalize all thisto 1 before plotting(P(n,v).

Then you record the responses (spike count n) for these stimuli. For example the red stimulus gives you after many repetitions the red responsecurve.

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Neuronal Codes – Probabilistic response and Bayes’ rule

Summing all values along the red arrow yields P(n) the Prior how often a certain number of spikes in general is observed .

With Bayes and theknowledge of P(n) we canget the two conditionalprobability curves, too.

With Bayes and theknowledge of P(n) we canget the two conditionalprobability curves, too.

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Neuronal Codes – Probabilistic response and Bayes’ rule

average number of spikes

depending on stimulus velocity

average stimulus depending on

spike count

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Neuronal Codes – Probabilistic response and Bayes’ rule

average number of spikes

depending on stimulus velocity

average stimulus depending on

spike count

non-linearrelation

almost perfectly linearrelation

The left relation is MUCH easier to understand than the right one (which isthe one you would have measured naively)!

This is how Bayes can help. You can deduct (guess) the stimulus velocity(here linearly) from just the spike count.

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Neuronal Codes – Probabilistic response and Bayes’ rule

For a deeper discussion read, for instance, that nice, difficult book:

Rieke, F. et al. (1996). Spikes: Exploring the neural code. MIT Press.