[Problem Set 2 Quantitative models of behavior] Problem 3: The drift diffusion model of decision-making
Link of the iPython notebook for the code
AT2 – Neuromodeling: Problem set #2 QUANTITATIVE MODELS OF BEHAVIOR
PROBLEM 3: The drift diffusion model of decision-making.
We consider a two-alternative forced choice task (2AFC-task): the subject (e.g. a monkey) sees a cluster of moving dots (in many directions) on a screen and is to choose (whenever he/she/it wants) whether they are moving upwards or downwards.
In the drift-diffusion-model, the subject is assumed to compare two firing rates:
- one firing rate of an upward-motion sensitive neuron, denoted by
- and another one from a downward-motion sensitive neuron, denoted by
Then, the subjects integrates the difference as follows:
where
For a given threshold
- if
then the subject chooses - else if
then the subject chooses
We have the following discrete approximation of the drift-diffusion-model:
1. Reaction times
Let
Let us run the drift-diffusion-model ten times with the above parameters, with resort to the Euler method:

Now, after running the model
-
For
:Figure 1.1. - Distribution of reaction times for outcome and after running the model times -
For
:Figure 1.2. - Distribution of reaction times for outcome and after running the model times
2. Probability of outcome
Let us define the evidence for outcome
The analytical formula of the probability of outcome
where
Let us compare it, for values of
The problem is that we have to compute this empirical probability for several values of
But we can cope with that with a trick!
Fast algorithm to compute the distribution of reaction times
From the discrete approximation of the drift-diffusion-model:
So for all
where the
are independent and normally distributed random variables.
But it is well known that a sum of independent and normally distributed random variables is also a normally distributed variable, such that:
i.e.
But as it happens:
So for all
where
So solving for
is tantamount to:
-
Analysis: solving (for
, in ) two quadratic equations for a given : -
Synthesis: keeping only the roots
such that , and setting
which gives us a very fast algorithm!

Finally, we can compare the empirical probability of outcome


We see that the empirical probability of outcome
-
as
increases -
as
decreases (with an almost perfect match above for )
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