Why Biological Systems Suddenly Change State: An Intuitive Guide to Freidlin–Wentzell Theory

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  Stochasticity is ubiquitous in biology and neuroscience, manifesting in various forms, including ion channel noise, synaptic variability, gene regulatory fluctuations, noisy population dynamics, and more. Many biological systems spend long periods in a stable “state” and only rarely transition to another state due to noise. For instance, a neuron typically remains inactive but may occasionally trigger a spontaneous spike. Similarly, a gene can switch from the OFF state to the ON state due to rare bursts of transcription factors. Cells can also transition out of metabolic or epigenetic states, populations might shift between different ecological equilibria, and a viral infection can fluctuate between phases of control and uncontrollability. Freidlin–Wentzell theory provides a mathematically rigorous framework to study these phenomena when noise is small but nonzero . It tells you, firstly, h ow likely rare transitions are,    secondly,   h ow fast they occ...
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The Drift Diffusion Model (DDM) for Decision Making in the TAFC task

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In applying the diffusion model to the TAFC, it is assumed that the accrual of noisy evidence corresponding to the two alternatives (e1, e2) is carried on until their difference (e1–e2) reaches a decisional threshold at the upper value (Th) or at the lower value 0. The attainment of one of these critical values indicates where the preference is directed: the upper threshold relates to the positive sign of the difference (e1–e2), while the lower thresholds corresponds to the negative value of (e1–e2). 
The time necessary to reach one of the boundaries, i.e. the response time RT, depends on:

a) the distance between the boundaries and the starting point; 
b) the drift, i.e., the rate at which the average (trend) of the random variable (e1–e2) changes; 
c) the diffusion, i.e., the variability of the path from the trend (Figure 1). 

These elements characterizes the so called drift diffusion model (DDM). The accumulation of evidence is driven both by a deterministic component (drift) that is proportional to the stimulus intensity and by a stochastic component of noise that makes the evidence deviate from its own trend. The variance of the noise is the diffusion parameter of the model. 

The rationale of DDM is that since the transmission and codification of the stimuli are inherently noisy, the quality of the feature extraction from such inputs may call for  accumulation of a sufficient large sequence of the stimuli to get information [1]. By knowing the threshold level and the RT enables one to take a sight into the mechanism underlying the decision process [2,3]. 

Figure1. The randomness of the path taken under the influence of noisy stimuli characterizes the  diffusion models. A stimulus is represented in a diffusion equation by its influence on the drift rate of a random variable. This random variable, say the difference of evidence corresponding to the alternatives, accumulates the effects of the inputs over time until one of the boundaries is reached. The decision process ends when evidence reaches the threshold and the time at which it occurs is called response time (RT). Therefore, the drift term represents the weight of evidence in favor of one alternative. The variance of noise in the input signals determines the  diffusion of the path of the random variable. We can draw an analogy with a physical system and imagine the decisional process as the state of a “particle” moving within a potential well. Under this point of view, the persistence for relatively long periods of the state variable in the sub-threshold area implies that the particle still entangled in the potential well, enters an excited state where it remains for an exponentially distributed time interval with a certain decay time τd. If the combination of input and noise is sufficiently strong, then the particle is able to jump the barrier, i.e. the threshold, and the system returns to an equilibrium state. The dynamics of the particle thus may resolve in a relaxation process [4] characterized by the oscillations between periods of sub-threshold “disorder” inside the potential well and short impulses that trigger the system beyond the threshold in the rest state. This physical analogy allows to better perceive how the DDM may fit the evolution of the input-output map underlying the neuronal model of the decision-making process.

It has been shown [5,6] that under experiments with human subjects performing TAFC tasks, the DDM yields accuracy and reaction times (RTs). Moreover, the RTs estimated by the DDM tend to distribute as an asymmetric random variable, that is, so as it would be expected from human performance, because the RT distributions are usually skewed toward longer times. An advantage from DDM is that, given a level of accuracy, it results the fastest decision maker, for a fixed decision threshold. 

The noisy signals emphasize the role of the thresholds, in fact, thresholds have no effect on accuracy if the noise is absent. Instead, the noise makes the variable representing the difference of evidence corresponding to the alternatives, stochastic. The fluctuations of this random variable generates series of erroneous responses, and the response times refer both to the correct and to error responses. The accuracy tends to increase proportionally to the rising of threshold which results in a speed–and-accuracy tradeoff. 

The speed-and-accuracy tradeoffs are usually reproduced by adjusting the boundaries such that lower thresholds produce faster but less accurate responding, whereas higher thresholds produce more accurate but longer decision times in order to average out uncorrelated noise. This speed–accuracy trade-off is usually considered a basic parameter for interpreting the results both of behavioral experiments and, as before reported, neurological experiments [2,7,8]. However, the surprising capability of DDM to fit behavioral and neurological data seems to indicate that some decision making process in the brain are really computed by a similar mechanism that accumulates evidence [9].
  1. Ratcliff, R., Smith, P.L. (2004). A comparison of sequential sampling models for two-choice reaction time. Psychol Rev. 111:333-367.
  2. Shadlen, M.N., Hanks, T.D., Churchland, A.K., Kiani, R., Yang, T. (2007). The speed and accuracy of a simple perceptual decision: a mathematical primer. In Doya, K., Ishii, S., Pouget, A., Rao, R.P.N. (Eds.) Bayesian Brain: probabilistic approaches to neural coding. The MIT Press, Cambridge (USA).
  3. Voss A., Rothermund K., Voss J. (2004). Interpreting the parameters of the diffusion model: an empirical validation. Memory&Cognition, 32(7), 1206-1220.
  4. Pikovsky A., Rosenblum M., Kurths J. (2001). Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, USA.
  5. Ratcliff, R. (1978). A theory of memory retrieval. Psychol Rev. 85, 59-108.
  6. Laming, D. R. J. (1968). Information theory of choice-reaction times. Wiley, NewYork.
  7. Roitman, J.D., Shadlen, MN. (2002). Response of neurons in the lateral intraparietal area during a combined visual discrimination reaction time task. J Neurosci. 22: 9475-9489. 
  8. Wickelgren, W. A. (1977). Speed–accuracy tradeoff and information processing dynamics. Acta Psychol 41: 67–85.
  9. Deco, G., Rolls, E.T., Romo, R. (2009). Stochastic dynamics as a principle of brain function. Prog Neurobiol. 88: 1-16.



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