J Comput Neurosci DOI 10.1007/s10827-013-0465-5 Bifurcation study of a neural field competition model with an application to perceptual switching in motion integration J. Rankin · A. I. Meso · G. S. Masson · O. Faugeras · P. Kornprobst Received: 25 January 2013 / Revised: 19 May 2013 / Accepted: 20 May 2013 © The Author(s) 2013. This article is published with open access at SpringerLink.com Abstract Perceptual multistability is a phenomenon in broad range of perceptual competition problems in which which alternate interpretations of a fixed stimulus are per- spatial interactions play a role. ceived intermittently. Although correlates between activity in specific cortical areas and perception have been found, Keywords Multistability · Competition · Perception · the complex patterns of activity and the underlying mecha- Neural fields · Bifurcation · Motion nisms that gate multistable perception are little understood. Here, we present a neural field competition model in which competing states are represented in a continuous feature 1 Introduction space. Bifurcation analysis is used to describe the differ- ent types of complex spatio-temporal dynamics produced Perception can evolve dynamically for fixed sensory inputs by the model in terms of several parameters and for differ- and so-called multistable stimuli have been the attention ent inputs. The dynamics of the model was then compared of much recent experimental and computational investiga- to human perception investigated psychophysically during tion. The focus of many modelling studies has been to long presentations of an ambiguous, multistable motion pat- reproduce the switching behaviour observed in psychophys- tern known as the barberpole illusion. In order to do this, ical experiment and provide insight into the underlying the model is operated in a parameter range where known mechanisms (Laing and Chow 2002; Freeman 2005; Kim physiological response properties are reproduced whilst also et al. 2006; Shpiro et al. 2007; Moreno-Bote et al. 2007; working close to bifurcation. The model accounts for char- Borisyuk et al. 2009; Ashwin and Lavric 2010). Bifurcation acteristic behaviour from the psychophysical experiments analysis (Strogatz 1994; Kuznetsov 1998) and numerical in terms of the type of switching observed and changes in continuation (Krauskopf et al. 2007) are powerful tools the rate of switching with respect to contrast. In this way, from the study of dynamical systems that have already the modelling study sheds light on the underlying mecha- proved effective in analysing rate models where the com- nisms that drive perceptual switching in different contrast peting perceptual states are represented by discrete neural regimes. The general approach presented is applicable to a masses (Shpiro et al. 2007; Curtu et al. 2008; Theodoni et al. 2011b). Two commonly proposed mechanisms that drive the switching behaviour in these models are adapta- Action Editor: J. Rinzel tion and noise and a strong argument is made in Shpiro J. Rankin () · O. Faugeras · P. Kornprobst et al. (2009) that a balance accounts best for experimental Neuromathcomp Team, Inria Sophia Antipolis, findings across different model architectures and different 2004 Route des Lucioles-BP 93, Alpes-Maritimes, 06902, France adaptation mechanisms. Existing studies using discrete neu- e-mail: [email protected] ral masses have shown that switching in rivalry experiments can be described by a relatively simple dynamical system. · A. I. Meso G. S. Masson However, the problem of multistable motion integration is Institut de Neurosciences de la Timone, CNRS et Aix-Marseille Universit´e, Campus Sant´e Timone, different because the perceived direction of motion is rep- 27 Bd Jean Moulin, Marseille 13385, France resented on a continuous scale. We therefore asked the J Comput Neurosci following questions. Can a minimal model with a contin- Meso et al. 2012b). During extended presentations of these uous feature space describe switching behaviour in motion stimuli, the dominant percept switches randomly and the integration? Do qualitative changes in the dynamics pre- dominance durations between switches have been shown dicted with bifurcation analysis correspond to changes in to fit certain distributions dependent on the experimen- the mechanisms driving the switches? tal paradigm (Levelt 1968; Leopold and Logothetis 1996; Here, we will take advantage of the neural fields formal- Logothetis et al. 1996;Lehky1995; Zhou et al. 2004; Rubin ism (Amari 1971; Wilson and Cowan 1972, 1973)inorder and Hup´e 2005). to study neural competition in a model with a continuous Here, we will study the temporal dynamics of perception feature space where adaptation and noise are implemented for the so-called multistable barberpole illusion, which has as mechanisms that can drive activity switches. The model been investigated in complementary psychophysical exper- describes the mean firing rate of a population of feature iments (Meso et al. 2012b). Some of these results will be selective neurons. Deterministic versions of this feature- presented alongside the modelling work. We will demon- only model with spike frequency adaptation have been strate how the general neural fields model can reproduce the studied previously without input (Curtu and Ermentrout main dynamical characteristics of the perceptual switches 2004) and with a unimodal input (Hansel and Sompolinsky observed in the experiments. We use the mean switching 1998; Folias 2011). A key difference with existing rivalry rates reported in the experiments to constrain model param- models is that the competing percepts form tuned responses eters and propose specific mechanisms that can account for in a continuous feature space instead of being repre- the behaviour in different contrast regimes. Importantly, we sented by discrete populations as in, for example, Shpiro will show that the two contrast regimes identified experi- et al. (2007), and Theodoni et al. (2011b). The more gen- mentally, one in which the rate increases with contrast, the eral model we use allows for perceptual transitions to occur other in which the rate decreases with contrast, are linked in a smooth way as opposed to discrete switches between to specific mechanisms with the model. Although a com- two isolated percepts. Starting from the results presented bination of noise and adaptation drive the switching, the in Curtu and Ermentrout (2004), we will first introduce dominant mechanism changes with contrast. Furthermore, a simple (unimodal) input and investigate how the vari- we are able to quantify this in an experimentally testable ous types of solutions from the no-input case are modified. way: the distribution of dominance durations fit different With the application of numerical bifurcation methods we statistical distributions in each contrast regime. find that although the boundaries between parameter regions In Section 2 section we give a mathematical descrip- featuring different types of responses are gradually dis- tion of the model before presenting general results that map torted with increasing input strength, much of the global the model’s possible behaviours across parameter space in structure is preserved. This allows for all possible types of Section 3 and then applying the model to the study of behaviour, and parameter regions for which it can occur, multistable perception in Sections 4 and 5. to be comprehensively described across a wide range of model parameters controlling input gain, adaptation gain and the shape of the firing rate function. For a simple input 2 Competition model with continuous feature space we are able to match the models output to known response properties from the literature before considering the intro- 2.1 The neural field framework duction of a complex (multimodal) input that gives rise to multistable behaviour. The neural field equations provide an established frame- In this paper we are interested in moving, ambiguous work for studying the dynamics of cortical activity, repre- visual stimuli for which two or more distinct interpretations sented as an average membrane potential or mean firing are possible, but where only one of these interpretations, rate, over a spatially continuous domain. Since the semi- or percepts, can be held at a time. Not only can the ini- nal work by Amari (1971), Wilson and Cowan (1972)and tial percept be different from one short presentation to Wilson and Cowan (1973) a broad range of mathemati- the next, but for extended presentations, the percept can cal tools have been developed for their study, see reviews change, or switch, dynamically. This phenomenon of multi- by Ermentrout (1998), Coombes (2005) and Bressloff stability has been observed and investigated with a number (2012) along with Ermentrout and Terman (2010, Chap- of different experimental paradigms, e.g. binocular rivalry ter 11) for a derivation of the equations. The equations experiments (Levelt 1968;Blake1989, 2001), apparent describe the dynamical evolution of activity of one or motion (Ramachandran and Anstis 1983), motion plaids more connected populations of neurons, each defined in that are bistable (Hup´e and Rubin 2003) or tristable (Hup´e terms of a spatial domain that can represent either physical and Pressnitzer 2012) and the multistable barberpole space (on the cortex), an abstracted feature space (ori- illusion (Castet et al. 1999; Fisher and Zanker 2001; entation, direction of motion, texture preference, etc.), or J Comput Neurosci some combination