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Erklären Sie den Unterschied zwischen P(t,s), P(s,t), P(t|s) und P(s|t). Gibt es hier Gleichheiten zwischen irgendwelchen dieser Terme?

Wann gilt P(s,t) = P(s) x P(t)?

Erklären Sie die nachfolgende Formel intuitiv: )()(|}{)(},{ tsPtstPtstP ii

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

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

what we see:

what ourbrain “sees”:

Erklären Sie den Unterschied zwischen „We“ und „Brain“ view hier. Beide sehen P(t,s) [linke Seite] aber beide sehen dies auf verschiedene Weise [rechte Seite]. Wieso?

Wie mißt man P(t,s) direkt?

Wie mißt man P(s,t) direkt?

Vervollständigen Sie die Bayes Formel für: P(s|t)= ????

Warum heißt die „reverse Korrelation“ so?32

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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 having presented 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 the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".

Joint probability is the probability of two events in conjunction. That is, it is the probability of both events together. The joint probability of A and B is 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 the dice are rolled, without any knowledge of the roll's outcome. For example, die 1 is equally 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 joint probability 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 that the 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 the probability 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, this has no impact on the probability of event B, which only depends on dice 2. We say events A and B are statistically independent or just independent and in this case: P(B|A)=P(B).

)(},{ tstP iJoint

Conditional P[f tigjs(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

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Neuronal Codes – Probabilistic response and Bayes’ rule

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

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

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

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

iii tP

tsPtstPttsP

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

)(|}{ tstP i

determine probability ofa spike trainfrom a given stimulus

How to measure some distributions ?

Here: The standard way!

Stimulus Response

Raster Plot

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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

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

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

iii tP

tsPtstPttsP 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:

Average spike count n dependent on the stimulus (velocity v).

Stimulus Response (as before)

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 than others , which makes this peaked.

Do this for all stimuli and 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 response curve.

Comes from motion sensitive neuron H1 in the fly’s brain:

Neuronal Codes – Probabilistic response and Bayes’ rule

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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 the knowledge of P(n) and P(v) we can get the two conditional probability curves, too.

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Neuronal Codes – Probabilistic response and Bayes’ rule

Easy to measure (Raster Plot)

Easy to measure (but takes long)

Often difficult to measure directly(use Bayes instead)

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

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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-linear relation

almost perfectly linearrelation

The left relation is MUCH easier to understand than the right one (which is the 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

)(|}{ tstP i

determine probability ofa spike trainfrom a given stimulus

Remember ?

Measuring the spike train given one stimulus is easy. Also measuring all spike trains using all kinds of stimuli (takes long though)

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From P(s,t) and P(t,s) and P(t), so far we had then used Bayes to determine P(t,s).

Can we also measure this ?

Measuring P(s,t) was easy see raster plot before, but how is measuring P(t,s) done?

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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

Called „Reverse Correlation“ as we look always back from a spike in time asking which stimulus chunck was (just) before the spike (and could have, thus, driven the given spike).

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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.