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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MODELS OF BASAL GANGLIA IN SACCADIC MOVEMENTS

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The eye movement system is closely related to cognitive functions such as perception, attention and memory. This is not surprising since eye movements provide the easiest and the most accurate way to extract information from our visual environment and the eye movement system largely determines what information is selected for further processing. This would make the saccadic system an ideal model for studying the decision-making (DM) process [1]. In general, the study of DM has been carried out under three main approaches, that involve economics, cognitive sciences and neurobiology. They differentiate on the base of their growing level of complexity [2]. Neuroeconomics is a relatively recent field of interdisciplinary research that integrates those three approaches. The application of the methodologies and the results that typically own to the cognitive neuroscience, in order to build models grounded in neurobiology is what characterizes the novelty of the neuroeconomics. Thus, the implementation of the neuroeconomic models implies to search for the neurobiological bases of the processes that run during the execution of the “economical” tasks [3]. 

 The decisions about where and when to look are important because they not only provide insight into the cognitive processes that occur within the brain, but also allow researchers to examine the physiological events that take place at a mechanistic level [4,5]. 
Saccades are quick, high frequent eye movements that address the fovea onto the visual targets. Many areas of the brain participate to the saccade generation: frontal cortical areas, lateral intraparietal cortex (LIP), BG, superior colliculus (SC) and the brainstem reticular formation.  As before described, the BG consist of brain nuclei that are supposed to be involved in several motor learning and function. They receive afferent connections from the cortex through the striatum and return projections to the cortex through the thalamus. Since the BG form a reinforcement learning circuit that uses reward (i.e., positive reinforcing) information from the environment, they facilitate those actions to which correspond the largest (probability of) reward. A firstly family of models for the saccade generation [6] operates by comparing the desired eye position to an internal model of location. The error estimated from this comparison is then used to correct the eye position towards the desired target. Because the SC receives projections from many cortical areas (both visual and cognitive centers) it is supposed to play a major role in the integration of information coming from different pathways in the brain for the saccadic control. More explicitly, the TD error signal would be given by the dopamine released by the mesencephalic nuclei such as the ventral tegmental area and substantia nigra pars compacta (SNc).
Reward information conveyed by dopamine release to BG nuclei switches striato-pallidal transmission between the direct and indirect pathways to BG [7]. The direct pathway involves the inhibitory GABA-ergic projections from the striatum (i.e., the input layer of BG that includes caudate nucleus and putamen) to the neurons of the SNr (i.e., the output layer of the BG). Striatal activation inhibits neurons in SNr, which in turn disinhibits the SC. The indirect pathway is formed by projections from striatum to SNr through the STN-GPe loop.
The BG contribution to the saccade generation, ultimately, presents three forms: 1) representation of error in dopamine signal; 2) use of dopamine signal for learning the saccade task; 3) the exploratory influence of the indirect pathway then enables moving away of the eyes from the current point of fixation. It was reported an inverse relation between BG output and saccadic accuracy (search efficiency), while reaction times are expected to be longer as BG activity diminishes (this is coherent with the results from the Parkinson’s Disease patients [7]). According to this model, the BG during (voluntary) saccadic movements gives rise to a reinforcement learning of the saliency map through both the direct pathway that performs exploitation and the indirect pathway that provides the exploration [8]. The dopamine signal, which serves as error signal, switches the onward transmission of cortico-striatal signals between the two pathways. 
  1. Trommershäuser, J., Glimcher, P.W., Gegenfurtner, K.R. (2009). Visual processing, learning and feedback in the primate eye movement system. Trends Neurosci. 32(11), 583-590.
  2. Glimcher, P.W., Fehr, E., Camerer, C., Poldrack, R.A. (Eds.) (2009). Neuroeconomics: decision making and the brain. Academic Press, USA.
  3. Glimcher P.W., Rustichini A. (2004). Neuroeconomics: the Consilience of Brain and Decision, Science 306: 447-452.
  4. Glimcher, P. W. (2001). Making choices: The neurophysiology of visual-saccadic decision making. Trends in Neurosciences, 24(11), 654-659. 
  5. Glimcher, P. W. (2003a). The neurobiology of visual-saccadic decision making. Annual Review of Neuroscience, 26, 133-179.
  6. Robinson, D. (1975). Oculomotor control signals. In Lennerstrand, G. & Rita, P.B. (Eds), Basic mechanisms of ocular motility and their clinical implications. Pergamon, Oxford, UK. 
  7. Krishnan, R., Ratnadurai, S., Subramanian, D., Chakravarthy, S. (2010). A model of basal ganglia in saccade generation. ICANN 2010, Part 1, LNCS 6352: 282-290.
  8. Devarajan, S., Prashanth, P.S., Chakravarthy, V.S. (2006). The Role of the Basal Ganglia in Exploration in a Neural Model based on Reinforcement Learning.  Int J Neural Syst. 16(2): 111-124.

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