Abstract
Many perceptual and cognitive processes, like decision-making and bistable perception, involve multistable phenomena under the influence of noise. The role of noise in a multistable neurodynamical system can be formally treated within the Fokker–Planck framework. Nevertheless, because of the underlying nonlinearities, one usually considers numerical simulations of the stochastic differential equations describing the original system, which are time consuming. An alternative analytical approach involves the derivation of reduced deterministic differential equations for the moments of the distribution of the activity of the neuronal populations. The study of the reduced deterministic system avoids time consuming computations associated with the need to average over many trials. We apply this technique to describe multistable phenomena. We show that increasing the noise amplitude results in a shifting of the bifurcation structure of the system.
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References
Amit D, Brunel N (1997) Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cereb Cortex 7:237–252
Attneave F (1971) Multistability in perception. Sci Am 225:63–71
Brody C, Romo R, Kepecs A (2003) Basic mechanisms for graded persistent activity: discrete attractors. continuous attractors, and dynamic representations. Curr Opin Neurobiol 13:204–211
Brunel N, Wang X (2001) Effects of neuromodulation in a cortical networks model of object working memory dominated by recurrent inhibition. J Comput Neurosci 11:63–85
Camera GL, Rauch A, Luescher H, Senn W, Fusi S (2004) Minimal models of adapted neuronal response to in vivo-like input currents. Neural Comput 16:2101–2124
Deco G, Rolls E (2006) Decision-making and weber’s law: a neurophysiological model. Eur J Neurosci 24:901–916
Glimcher PW (2003) Decisions, uncertainty, and the brain. MIT, Cambridge
Glimcher PW (2005) Indeterminacy in brain and behavior. Annu Rev Psychol 56 (NIL):25–56
Gold JI, Shadlen MN (2000) Representation of a perceptual decision in developing oculomotor commands. Nature 404:390–394
Gold JI, Shadlen MN (2002) Banburismus and the brain: decoding the relationship between sensory stimuli, decisions, and reward. Neuron 36(2):299–308
Laing C, Chow C (2002) A spiking neural model of binocular rivalry. J Computa Neurosci 12:39–53
Leopold D, Logothetis N (1999) Multistable phenomena: changing views in perception. Trends Cogn Sci 3:254–264
Machens C, Romo R, Brody C (2005) Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science 307:1121–1124
Mattia M, Giudice PD (2002) Population dynamics of interacting spiking neurons. Phys Rev E 66(5):051917
Mattia M, Giudice PD (2004) Finite-size dynamics of inhibitory and excitatory interacting spiking neurons. Phys Rev E 70:052903
Platt ML, Glimcher PW (1999) Neural correlates of decision variables in parietal cortex. Nature 400(6741):233–238
Ratcliff R, Zandt TV, McKoon G (1999) Connectionist and diffusion models of reaction time. Psychol Rev 106(2):261–300
Renart A, Brunel N, Wang XJ (2003) Mean field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. In: Feng J (ed) Computational neuroscience: a comprehensive approach. Chapman and Hall, Boca Raton, pp 431–490
Rodriguez R, Tuckwell HC (1996) Statistical properties of stochastic nonlinear dynamical models of single neurons and neural networks. Phys Rev E 54:5585–5590
Rodriguez R, Tuckwell HC (1998) Noisy spiking neurons and networks: useful approximations for firing probabilities and global behavior. BioSystems 48:187–194
Rolls ET, Deco G (2002) Computational neuroscience of vision. Oxford University Press, Oxford
Romo R, Salinas E (2001) Touch and go: Decision-making mechanisms in somatosensation. Annu Rev Neurosci 24:107–137
Romo R, Salinas E (2003) Flutter discrimination: neural codes, perception, memory and decision making. Nat Rev Neurosci 4:203–218
Romo R, Hernandez A, Zainos A (2004) Neuronal correlates of a perceptual decision in ventral premotor cortex. Neuron 41:165–173
Schall J (2001) Neural basis of deciding, choosing and acting. Nat Rev Neurosci 2:33–42
Shadlen MN, Newsome WT (1996) Motion perception: seeing and deciding. Proc Natl Acad Sci USA 93(2):628–633
Smith P, Ratcliff R (2004) Psychology and neurobiology of simple decisions. Trends Neurosci 23:161–168
Taylor M, Aldridge K (1974) Stochastic processes in reversing figure perception. Percept Psychophys 16:9–27
Thompson KG, Hanes DP, Bichot NP, Schall JD (1996) Perceptual and motor processing stages identified in the activity of macaque frontal eye field neurons during visual search. J Neurophysiol 76(6):4040–4055
Tuckwell HC (1988) Introduction to theoretical neurobiology. Cambridge University Press, Cambridge
Tuckwell HC, Rodriguez R (1998) Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks. J Comput Neurosci 5:91–113
Usher M, McClelland JL (2001) The time course of perceptual choice: the leaky, competing accumulator model. Psychol Rev 108(3):550–592
Wilson H (2003) Computational evidence for a rivalry hierarchy in vision. Proc Nat Acad Sci USA 100:14499–14503
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Deco, G., Martí, D. Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems. Biol Cybern 96, 487–496 (2007). https://doi.org/10.1007/s00422-007-0144-6
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DOI: https://doi.org/10.1007/s00422-007-0144-6