Bifurcation analysis applied to a model of motion integration with a multistable stimulus
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Abstract
A computational study into the motion perception dynamics of a multistable psychophysics stimulus is presented. A diagonally drifting grating viewed through a square aperture is perceived as moving in the actual grating direction or in line with the aperture edges (horizontally or vertically). The different percepts are the product of interplay between ambiguous contour cues and specific terminator cues. We present a dynamical model of motion integration that performs direction selection for such a stimulus and link the different percepts to coexisting steady states of the underlying equations. We apply the powerful tools of bifurcation analysis and numerical continuation to study changes to the model’s solution structure under the variation of parameters. Indeed, we apply these tools in a systematic way, taking into account biological and mathematical constraints, in order to fix model parameters. A region of parameter space is identified for which the model reproduces the qualitative behaviour observed in experiments. The temporal dynamics of motion integration are studied within this region; specifically, the effect of varying the stimulus gain is studied, which allows for qualitative predictions to be made.
Keywords
Motion Perception Multistability Visual cortex Barber pole Bifurcation1 Introduction
The interesting and longstudied aperture problem constitutes an important ambiguity that must be resolved by the visual system in order to attribute an accurate direction of motion to moving objects (Wallach 1935; Wuerger et al. 1996). The motion of a uniform contour is consistent with many possible directions in the absence of terminator information provided by line endings. The ambiguous contour information is referred to as a 1D cue and the specific terminator information, which can be intrinsic to the object or produced by an occluding aperture, is referred to as a 2D cue.
In this article, we focus on a classical psychophysics stimulus used to probe the interactions between 1D and 2D motion cues, the socalled “barber pole” illusion (Chapter 4 Hildreth 1983). A a diagonally drifting grating viewed through an elongated rectangular aperture is perceived as drifting in the direction of the long edge of the aperture. The illusion is generated by the larger number of unambiguous (but misleading) 2D cues parallel to the long edges of the aperture. Mitigating the effect of the 2D cues can break the illusion as demonstrated in experiments by introducing a depth separation between grating and aperture (Shimojo et al. 1989), or, by introducing indentations on the aperture edges (Kooi 1993). More interesting is the fact that for certain stimulus parameters, the barber pole can elicit a multistable perceptual response. The effect of the ratio of the lengths of the aperture edges, or terminator ratio, was investigated in Castet et al. (1999) and Fisher and Zanker (2001). For large ratios the direction of the long aperture edge dominates perception. However, for ratios close to 1:1 the stimulus is multistable with a trimodal response for the perceived direction; the main percepts either agree with the direction perpendicular to the grating’s orientation or with one of the two aperture edge directions. Another interesting aspect in resolving the aperture problem is the temporal dynamics of the integration of 1D and 2D cues. The barber pole illusion was studied in ocular following experiments (Masson et al. 2000); early tracking responses were shown to be initiated in the contour direction with a later response in the direction of the long axis of the aperture. The existence of an early response dominated by 1D cues, which is later refined when 2D cues are processed is supported by further studies in ocular following Barthélemy et al. (2010), psychophysics (Lorenceau et al. 1993) and physiology (Pack and Born 2001).
The middle temporal area (MT) of the visual pathway, that receives the majority of its synaptic inputs from the primary visual cortex (V1), plays a key role in the perception of moving objects and, more specifically, the solution of the aperture problem. MT is characterised by directionselective neurons that are organised in a columnar fashion, similar to the organisation of orientationselective neurons in V1 (Diogo et al. 2003). For an extensive discussion of the function of MT, see the following review articles (Britten 2003; Born and Bradley 2005). Cortical responses of MT have been linked specifically to perception of motion; see again the review article (Britten 2003) and, more recently, the paper (Serences and Boynton 2007).
Several models of motion integration have been proposed in the literature to solve the aperture problem, providing some insights into the underlying neural mechanisms. Building on the first linear/nonlinear models (Chey et al. 1997; Simoncelli and Heeger 1998), several approaches added extensions to modulate the motion integration stages: feedback between hierarchical layers (Grossberg et al. 2001; Bayerl and Neumann 2004), inclusion of input form cues (Berzhanskaya et al. 2007; Bayerl and Neumann 2007), luminance diffusion gating (Tlapale et al. 2010b), or depth cues (Beck and Neumann 2010). Although these models reproduce the predominant percepts in a wide range of stimuli, in none of the articles describing them are multistable results depicted. Furthermore, although they show some limited individual case of dynamical behaviour just on the level of simulations, there is no rigorous analysis of the dynamical behaviour and no comprehensive parameter studies that fully explore all possible dynamical behaviour. In summary, the questions of what mechanisms are behind multistable motion perception and what dynamical processes are involved have largely been overlooked from the modelling point of view.
The focus of this paper will be the analysis of a mathematical model of motion integration with an input that generates a multistable perceptual response. To do so, we study cortical behaviour at the population level; see Pouget et al. (2000) for a discussion of how information can be encoded at a population level. We work within the neural fields formalism, a mathematical framework that was originally studied in Wilson and Cowan (1972) and Amari (1971, 1972); see Chapter 11 Ermentrout and Terman (2010) for various derivations of the equations. Instead of looking at the spiking behaviour of individual, interconnected neurons, the neural field approximation represents the mean firing rate of a neural population at the continuum limit and activity levels are represented in a spatially continuous way. Neural fields equations have been successfully applied to the study of motion in, e.g., Giese (1998), Deco and Roland (2010) and Tlapale et al. (2010a). In the later, the complex model presented, describing behaviour of multiple cortical layers and their feedforward and feedback connections, was capable of performing motion integration on both natural image sequences and classical psychophysics presentations. In terms of multistable stimuli, the 1:1 barber pole discussed above lead to coexisting steady states in the model, but the temporal dynamics of multistable perception were not investigated. We aim to develop a tractable model of manageable complexity that allows for a detailed study of the temporal dynamics of multistable motion perception using powerful tools from dynamical systems theory.
A natural tool for the study of dynamical systems for which multiple steadystate solutions coexist is bifurcation analysis. Throughout the manuscript when the term solution is used this refers to a steadystate solution. In dynamical systems theory, a bifurcation is a critical point encountered under the variation of one or more parameters at which there is a change in the stability and number of solutions. Indeed, under the variation of a parameter a solution of a dynamical system will vary in state space; when the solution is plotted in terms of some norm against the parameter, it lies on a solution branch. A dynamical system may have multiple solution branches that are characterised by, for example, different spatial and stability properties. At special points when varying a parameter, solution branches can meet and bifurcate from each other. At these socalled bifurcation points, the number of solutions and their stability changes. In order to know what type of behaviour a model can produce it is necessary to gain full understanding of the type of bifurcations that occur, the types of solutions that are involved and at which parameter values. For a general introduction to bifurcation analysis of finite dimensional systems see Strogatz (1994) and Kuznetsov (1998), and of infinite dimensional systems with symmetries see Chossat and Lauterbach (2000) and Haragus and Iooss (2010). Bifurcation analysis has been used to study pattern formation in a number of different settings (Ermentrout and Cowan 1980; Bressloff and Kilpatrick 2008; Coombes and Owen 2005). More specifically, a spatialised model of V1 has been used to investigate hallucinatory visual patterns (Bressloff et al. 2001; Golubitsky et al. 2003; Bressloff and Kilpatrick 2008), localised patterns have been studied in models of working memory (Laing et al. 2002; Guo and Chow 2005; Faye et al. 2012) and in a model of texture perception (Faye et al. 2011). In all of these studies only spontaneous activity is studied, that is, in the absence of any cortical input.
In this article, we propose a spatialised ring model of direction selection, where the connectivity in the direction space and physical space is closely related to the Mexicanhat type connectivity typically used in the ring model of V1 (BenYishai et al. 1995; Somers et al. 1995; Hansel and Sompolinsky 1997). We apply analytical tools such as stability analysis and normal form computations in order to identify and categorise bifurcations in our model. These tools have been used successfully to study the neural field equations, see, for example Curtu and Ermentrout (2004), Coombes et al. (2007) and Roxina and Montbrióa (2011). However, in the presence of an input to the model, these analytical tools are no longer applicable; although certain perturbationtype methods can be applied if the input is considered to have a specific, simple spatial structure and to be small (Veltz and Faugeras 2010; Ermentrout et al. 2010; Kilpatrick and Ermentrout 2012). Note that in the study (Veltz and Faugeras 2010), large inputs with a simple spatial structure were also studied with numerical continuation. In this paper, we first investigate the model’s behaviour in the absence of an input using analytical techniques. Then building on the knowledge gained we apply the tool of numerical continuation to track solution branches under parameter variation and detect bifurcation points; effectively continuation provides a computational tool for performing bifurcation analysis. For an introduction to continuation algorithms see Krauskopf et al. (2007). For the problem we study here the continuation module LOCA of the numerical tools package Trilinos is well suited (Heroux et al. 2005). Numerical bifurcation tools have not previously been used to study a neural fields model in the presence of a large, spatially complex input. Furthermore, the application of bifurcation analysis and numerical continuation to the study of a model of motion integration is new. These methods allow us to build a complete picture of the model’s possible dynamics in terms of parameter regions exhibiting qualitatively different behaviour and to identify the boundaries between these regions. In this way, we are able to ensure that parameter regions in which a desired behaviour is present are not isolated and that the behaviour is robust with respect to small changes in the model set up. The typical approach of a simple numerical search in parameter space often misses important parameter regions and does not provide information about robustness of behaviour with respect to parameter variation. The technical parts of this paper form the basis of the strategy employed to fix model parameters based on biological and mathematical constraints. We perform a systematic study of the model’s solution structure depending on key parameters including the stimulus strength and identify a parameter region for which the model has steadystate solutions corresponding to the three predominant percepts (diagonal, horizontal, vertical) observed in experiment (Fisher and Zanker 2001). An extended study of the system’s temporal dynamics allows for experimental predictions to be made regarding the distributions of the different percepts seen for different length presentations. In particular we predict that for long enough presentations only the horizontal and vertical percepts should be seen.
2 Model of direction selection
The model described here uses a neural fields description of the firingrate activity of a population of neurons in middle temporal area (MT) over a physical (cortical) space and a feature space of motion direction. Two essential mechanisms are represented by the model: direction selectivity in the feature space and spatial diffusion of activity across the physical space. Stimulus input to the model is represented as preprocessed motion direction signals from V1 complex cells. Such a representation is comparable to classic motion detectors such as the output of elaborated Reichardt detectors (Van Santen and Sperling 1984; Bayerl and Neumann 2004) or of motion filters. The output of the model is the time evolution of activity levels across the physicaldirection space.
The functionality of the model is encoded by the connectivity across directionspace and physicalspace, which is processed by a nonlinearity. In the direction space, the connectivity is based on a narrowly tuned excitation with broadly tuned inhibition, as described for MT in Grunewald and Lankheet (1996). Such a Mexicanhattype connectivity in the orientationspace has been used previously for the ring model of orientation selection (BenYishai et al. 1995; Bressloff et al. 2001). Here, we assume that the inhibition tuning in MT is broad enough to be approximated by uniform lateral inhibition; with uniform lateral inhibition when cells selective for a particular direction are active, cells selective for the opposite direction are inhibited. This is consistent with known direction selectivity properties of MT (Albright 1984; Diogo et al. 2003) and with recordings from MT with transparent drifting dot patterns moving in opposite directions (Snowden et al. 1991). Furthermore, the uniform lateral inhibition connectivity has the convenient property of fixing the first nontrivial Fourier mode of the connectivity to be the largest, which is necessary for the model to produce tuningcurve solutions. In the physical space, diffusion is captured by an inverted Mexican hat connectivity, which has been used in a number of neural fields models with delays; see, for example, Hutt et al. (2003), Venkov et al. (2007) and Chapter 6 Veltz (2011). As motivated in Venkov et al. (2007), for cortical tissue the principal pyramidal cells are often surrounded by inhibitory interneurons and their long range connections are typically excitatory (Gilbert et al. 1996; Salin and Bullier 1995; McGuire et al. 1991). The inverted Mexican hat connectivity propagates activity outwards from stimulated regions and is consistent with a model output that describes a coherent motion across the physical space.
2.1 Representation of the stimulus
The inclusion of 2D physical space would allow for many variations to the stimulus to be considered. However, there are a number modifications that are possible with the present 1D representation that we now discuss. For example, rectangular apertures are often considered with a specific ratio between the different edges (Fisher and Zanker 2001) and by taking a 1D cut across the stimulus this information would be lost. However, a similar effect could be achieved by giving a stronger weighting to the longer aperture edge. Another feature often investigated in psychophysics experiments is the angle of the grating with respect to the aperture edges (Castet et al. 1999). Again, although certain information is lost in the 1D cut, such a change to the stimulus could be considered by placing the 2D elements as shown in Fig. 1(c) asymmetrically about v = 0 (but still orthogonal to each other). Furthermore, the choice of the cut made Fig. 1(b) is quite arbitrary. For example, moving the cut closer to the line between the top left and bottom right corners would widen the stimulated region in x; in terms of the models response, we would expect the further separation of the 2D cue elements to delay the time it takes for the 2D cues to affect on the dynamics. The specific case where a cut is taken between two corners of the aperture would result in there being competition between 2D different cues at the aperture edge points. The stimulus shown in Fig. 1(c) could be modified to incorporate this by adding points at both v = ±45° at each aperture edge x = ±0.75 as opposed to v = 45° at x = − 0.75 and v = − 45° at x = 0.75 as is currently the case. Overall the 1D approximation used here is well suited to the specific stimulus studied with its particular symmetry properties, that is, with a square aperture and a grating that forms an equal angle with each aperture edge.
2.2 Neural field equation
An additional linear threshold term ν _{3} p is used to tune the maximum firing rate of directionselected solutions, whereas the constant threshold T is used to tune the firing rate of homogeneous, nondirectionselected solutions. It is convenient to include the linear threshold term in the connectivity operator as this simplifies the stability computations in Section 3.2. The choice of all parameters is discussed in Section 4.
3 Analytic results in the absence of stimulus, k = 0
In this section we discuss inherent properties of Eq. (2) and its solutions without a stimulus input. The results we obtain analytically provide a foundation of knowledge about the different types of solution the model can produce, the role of key parameters and a means to set appropriate values of parameters based on mathematical and biological constraints.
We begin our study by looking at the symmetry properties satisfied by the connectivity and the governing equation in Section 3.1. We show that Eq. (2) with the connectivity J as described above is equivariant with respect to a certain symmetry group. This important property dictates the types of solution that can be produced by the model. Furthermore, it determines the type of bifurcations that occur.
In the single population model that we consider here, the only types of solutions that we encounter are steady states (or, persistent states). Given an initial condition p _{0}, the time evolution of the equations can be computed numerically; the particular initial condition chosen will determine which steadystate solution the system converges to. It is important to note that the transient dynamics encountered before the system converges to a steady state can also be greatly affected by the initial condition. In Section 3.2 we calculate analytically the steady states that have the additional property of being independent of both the physical and direction space. For these spatially independent solutions a constant level of activity persists across (x,v)space; this type of solution can be thought of as the baseline activity that we would see in the absence of a stimulus (or below the contrast threshold). We give an expression that allows us to compute these solutions depending on the system parameters. Also in Section 3.2, we compute the eigenvalues and eigenvectors of the connectivity operator J (a spatialmode decomposition), which allows us in turn to compute the stability of the steadystate solutions dependent on the nonlinearity stiffness λ. We show that for small enough λ the steadystate solutions are stable. As λ is increased, the most destabilising mode of J, as determined by its largest eigenvalue will become unstable at a critical value of λ. This critical value is the system’s principal bifurcation point.
We determine the type of the principal bifurcation in Section 3.3. Furthermore, given the mode of J that loses stability in this bifurcation, and given the symmetry properties of the governing equation, we are able to characterise the spatially dependent solutions produced by the model. A normal form computation determines the way in which the transition from spatially homogeneous solutions to spatially dependent solutions occurs in the model.
3.1 Symmetry group
Here we discuss the symmetry properties of Eq. (2), which will play an important role in determining the type of bifurcation that the model produces. The general concept is to specify the group of translations and reflections for which the governing equation is equivariant. The same group of translations and reflections, when applied to a solution of the equations, will produce coexisting solutions; for example, we will see in Section 3.3 that translational invariance in v means that a directionselected solution associated with one specific direction can be translated by any angle to give directionselected solutions associated with all other possible directions. Note that when a stimulus is introduced, the symmetry group of the equations will be in some way reduced and it is, therefore, important to first identify the full symmetry group before its introduction.
3.2 Spatially homogeneous solutions and their stability

The mode (0,0) is stable because ζ _{(0,0)} = ν _{1} − ν _{2} − ν _{3} < 0.

The \(\nu_2\,\hat{g}^I_x(j)\,\hat{b}^I_v(k)\) term ensures that all modes for which k = 0 are stable.

The positive \(\nu_1\hat{g}^E_x(j)\,\hat{g}^E_v(k)\) term produces the destabilising contribution, which is greatest for j = 0.

Further, this destabilising contribution is greatest for k closest to 0 and then diminishes for increasing k.
Parameter  Default value  Description 

μ  2  Decay rate 
T  −2  Constant threshold 
ν _{1}  3  Diffusion coefficient 
ν _{2}  66  Inhibition coefficient 
ν _{3}  1.5  Linear threshold 
\({\sigma}_{x}^{1}\)  0.5  Width of diffusion in x 
\({\sigma}_{x}^{2}\)  0.16  Width of inhibition in x 
\({\sigma}_{v}^{1}\)  0.16  Width of diffusion in v 
λ  Free parameter  Nonlinearity stiffness 
k  Free parameter  Input gain 
Bifurcation points associated with other modes that occur as λ is increased beyond λ _{ c } can be found in a similar fashion, however, it is the branch of solutions that are born from the principal bifurcation that will determine the types of spatially dependent solutions that the model will produce.
3.3 Normal form of the principal bifurcation point
In this section we classify the principal bifurcation point by first, applying the center manifold theorem and secondly, giving the appropriate change of variables to reduce the system’s dynamics into a normal form; we now introduce these concepts. In the previous section we performed a modal decomposition of the connectivity and computed the linear stability of Eq. (2) with respect to perturbations in the different modal components. For the different modal components, or eigenvectors χ _{(j,k)} (Eq. (16)), the sign of the associated eigenvalue \(\varrho_{(j,k)}\) (Eq. (17)) gives the linear stability. In the case when \(\varrho_{(j,k)}=0\) the stability is neutral; the parameter value for which this occurs is the bifurcation point. At this bifurcation point it is necessary to also consider nonlinear terms in some parameter neighbourhood in order to capture the local dynamics. A center manifold reduction allows us to compute these nonlinear terms by means of a leading order Taylor approximation; the center manifold theorem allows us to prove rigorously that the computed reduced system accurately captures the local dynamics. A normal form computation is a change of variables that classifies the type of bifurcation present in our system and allows for the dynamics local to the bifurcation point to be seen clearly. The coefficients found in the normal form computation provide important information about the direction of bifurcating branches in terms of the bifurcation parameter and the stability of these branches.
We prove in Appendix A that the relevant hypotheses for the centre manifold hold in our case. This computation depends both on the symmetry properties discussed in Section 3.1 and the fixed point stability analysis from Section 3.2. Indeed, in the previous section we identified the system’s principal bifurcation point as given by the pair \((\bar{p}_c,\lambda_c)\), solutions to the system (19). We now define respectively the first, second and third order coefficients in the Taylor expansion of S at \(\bar{p}_c\) to be S _{1}, S _{2} and S _{3}. We drop the subscript notation for the eigenvectors χ = e ^{ iv } and \(\overline{\chi}=e^{iv}\), which span the twodimensional eigenspace E _{1} associated with the eigenvalue ζ _{1}. The eigenvalues ζ _{0} (for the homogeneous mode) and ζ _{2} (for the j = 0, k = 2 mode) will also appear in the analysis that follows.

b < 0: The bifurcation is supercritical (Haragus and Iooss 2010). The new branch of solutions exist after the bifurcation, i.e. for λ > λ _{ c }. Furthermore the branch of solutions will be stable (attracting) local to the bifurcation. In our model, for increasing λ passing the bifurcation point, there would be smooth transition (smooth change in activity levels) from spatially homogeneous solutions to a directionselected solution.

b > 0 The bifurcation is subcritical (Haragus and Iooss 2010). The new branch of solutions exist before the bifurcation, i.e. for λ < λ _{ c }. Furthermore the branch of solutions will be unstable (repelling) local to the bifurcation. In our model, for increasing λ passing the bifurcation point, there would be noncontinuous transition (jump in activity levels) from spatially homogeneous solutions to a directionselected solution (see below for explanation).
 1.
In the absence of a stimulus there are no direction selected solutions;
 2.
When a stimulus is introduced a solution is selected that is intrinsically present in the model.
The bifurcating branch of solutions is characterised by the mode involved ((j,k) = (0,1)), that is, the solutions will be uniform in physical space and will have a single maximum in vspace; we can think of the maximal point as being centred at a selected direction. Hence, we refer to solutions on the bifurcated branch as directionselected solutions. Secondly, due to the O(2) symmetry in the absence of a stimulus, taking a directionselected solution, we know that it will still be a solution under any angular translation in v; i.e. there continuum (or ring) of solutions representing all possible selected directions.
4 Numerical results
In order to progress in the study of our model, we make use of computational tools. In Section 4.1 we take advantage of the analytical results from Section 3 in order to explore properties of the model dependent on parameters and to determine relevant ranges of the parameters. In Section 4.2 we build on the normal form computation from Section 3.3 by computing the bifurcated branch of solutions using numerical continuation. We compute the relationship between the activity levels along the bifurcated branch and certain parameters in order to fix their values. An in depth study of the solution structure of the model varying two parameters is given in Section 4.3. We identify a region of interest in parameters space, which we study in more detail in Section 4.4. In particular we investigate the temporal dynamics of the model which provides insight into the relation between the model’s solution structure and different stimulus driven responses.
The default parameter values that we use in Sections 4.3 and 4.4 are given in Table 1. We can arbitrarily set the decay parameter to μ = 2 and the diffusion parameter to ν _{1} = 3. Equally, these could be set to equal 1, however, the chosen values lead to all system parameters being of roughly the same order of magnitude, which facilitates the computations with Trilinos.
The activity levels for steady states of Eq. (2) are bounded by \([0,\frac{1}{\mu}]\). In order to simplify the presentation of the results it is convenient to give activity levels in terms of either the maxnorm \({p_{\textrm{max}}}\) or L _{2}norm p as a percentage of the maximum value \(\frac{1}{\mu}\). For the spatial connectivity, the surround width is set by \({\sigma}_{x}^{1}\) which is three times that of the centre width region \({\sigma}_{x}^{2}\) (Tadin et al. 2003). The spatial extent of the stimulus is roughly 1.5 times larger than the surround width. The width of the excitation in direction space is set by \({\sigma}_{v}^{1}\) to be about 20°. This relatively tight tuning allows for the model to produce direction selected solutions (tuning curves) of widths appropriate for distinguishing between the main percepts H, V and D; tuning widths are discussed further in Section 4.2. Finally, it remains to set values for T, ν _{2} and ν _{3}, which we do in Sections 4.1 and 4.2.
4.1 Parameter tuning
For a given value of ν _{2}, Fig. 4(a) provides the necessary range of T for which we have 5% \(<\bar{p}_c<20\)%. Furthermore, for values of (ν _{2},T) in this range, we always have the case b > 0 corresponding to a subcritical pitchfork bifurcation; this implies that it will be necessary to operate the model close to λ _{ f }. In order to set the final values of T, ν _{2} and ν _{3}, it is necessary to consider the level of activity and tuning width of the direction selected branch of solutions. In the next section we use the information presented here, along with the dependence of these solution properties on ν _{2} and ν _{3} to determine the final values of the parameters used.
4.2 Solution structure in the absence of stimulus
We now look at numerically computed solutions in the absence of stimulus (k = 0). In Section 3.3 it was shown that the system’s principal bifurcation is a pitchfork with O(2)symmetry; furthermore, it was shown in the previous section that this pitchfork bifurcation is subcritical for the parameter values specified in Table 1. We now employ numerical continuation to compute directly the bifurcated solution branch under the variation of λ. We use the continuation package LOCA, part of the Trilinos package of numerical tools. We use a discretisation of 37 points for both the physical space x and direction space v, verifying that the integrals involved in the solution of the equations satisfy suitable error bounds; such a discretisation gives rise to a system of 1369 ordinary differential equations (ODEs), which resulted in manageable computations with Trilinos. Increasing the discretisation further results in a drastic increase in computation times.
In order to set values of the parameters ν _{2} and ν _{3} we investigate how certain properties of the directionselected solutions change with respect to these parameters. Firstly, the solutions should have a suitable tuning width in vspace and, secondly, the activity should not be close to saturation. Given that we will operate the model at a value of λ close to λ _{ f }, it is convenient to study these properties at the fold point F _{1}, as shown in Fig. 5(a), and how they change under variation of the ν _{2} and ν _{3}. Note that as we vary either ν _{2} or ν _{3}, the value of λ for which the fold bifurcation occurs also changes. Therefore, we perform twoparameter continuation in λ and k, varying one parameter to satisfy the condition that the system be at a steady state and a second parameter to satisfy the condition that the system also be at a fold bifurcation.
Overall, in the last two sections, we have explored the relationship between the three parameters T, ν _{2} and ν _{3} and properties of the homogeneous and directionselected solutions. This has allowed us to set suitable values of these parameters such that the solutions produced by the model satisfy important biological and mathematical constraints; refer back to Table 1 for the parameter values used in the subsequent sections.
4.3 Introduction of stimulus and twoparameter analysis
From the results presented in the previous section we know that the sigmoidal slope λ should be set at a value for which the model cannot spontaneously produce directionselected solutions. This gives rise to the requirement that λ < 15.4 so as to be at a λvalue less than the first fold point labelled F _{1} in Fig. 5. We now investigate the solutions when a stimulus is introduced to the model by increasing the stimulus gain parameter k from 0; this is analogous to increasing the stimulus contrast. Recall that the stimulus described in Section 1 has the form shown in Fig. 1(c) when represented in the (x,v)plane.
At λ = 12.5, as shown in panel (a), there is a single branch of stable solutions on which no bifurcations are encountered. The solutions effectively mimic the input stimulus along the solution branch and the level of activity increases with k.
At λ = 13, for increasing k bifurcations are encountered at the fold points labelled \(F^H_1\) and \(F^V_1\), the subcritical pitchfork P _{1} and the supercritical pitchfork P _{2}. With the introduction of the stimulus, the O(2)symmetry of the pitchfork for k = 0, as discussed in the previous section, is broken. We now find that the pitchfork bifurcation at P _{1} gives rise to a pair of directionselectedsolution branches, as opposed to a continuum of solutions at all possible directions when k = 0 as discussed in Section 4.2. The two branches are associated with the directions H and V and the branches undergo a fold bifurcations \(F^H_1\) and \(F^V_1\), respectively. These two fold points coincide when plotted in terms of the solution norm p. The pair of directionselected branches reconnect at the supercritical (for k decreasing) pitchfork P _{2}. The main solution branch (that mimics the input stimulus and has a lower level of activity than the bifurcated branch) is unstable between P _{1} and P _{2}; it coexists with the directionselected solutions.
4.4 Temporal dynamics
By means of an indepth, twoparameter bifurcation analysis we have identified a region of interest in parameter space for which the model produces two stable directionselected solutions corresponding to the horizontal (H) and vertical (V) percepts, and one unstable directionselected solution corresponding to the diagonal (D) percept. We know that in this parameter region the system will converge to one of the stable percepts; in this section we study how the unstable solution D plays a role in the temporal dynamics. At the chosen parameter values in Table 1 with λ = 14, the model produces a low level of homogeneous activity in the absence of stimulus k = 0. We take this homogeneous state as an initial condition for simulations with a random perturbation drawn from a standard uniform distribution in order to introduce a stochastic element. The equations remain deterministic, but we introduce variability in the initial conditions for each simulation. First, we present an example simulation and define some quantities that characterise the temporal dynamics; next, we study how the temporal dynamics change with respect to the strength of the input gain k. The reader may find it helpful to refer back to the bifurcation diagrams Fig. 7(c) and (d), which show the solution branches corresponding to H, D and V.
5 Discussion
In this paper we presented a spatialised model of direction selection that was used to study the dynamical behaviour of multistable responses to the 1:1 ratio barber pole (a diagonally drifting grating viewed through a square aperture). The aim was to reproduce, at the cortical level, firing rate activity that can be related to qualitative behaviour observed in psychophysics experiments: the stimulus exhibits multistability of perception for short presentations where the dominant percepts are either (1) the diagonal direction of the grating (D) (2) in agreement with horizontal (H) aperture edge, or (3) in agreement with the vertical (V) aperture edge (Castet et al. 1999; Fisher and Zanker 2001). The model reproduced this multistable behaviour, where the percepts H and V correspond to stable solutions and the percept D to an unstable solution. The temporal dynamics were investigated and it was shown that early responses were dominated by the diagonal percept, midterm responses were trimodal between the three percepts and later responses were dominated by the two stable percepts H and V. This behaviour is consistent with experimental findings that show an early response dominated by 1D cues, which is later refined by 2D cues (Barthélemy et al. 2010; Lorenceau et al. 1993; Pack and Born 2001). One of the main predictions to be made from these results is that the percept D is only seen as a transient behaviour; for long term presentations on the order of seconds either H or V will be seen or perceived.
One of the main advantages of the model used here is its simplicity; the philosophy was to reproduce interesting behaviour observed in experiments with a minimalistic set of features to perform motion integration. Given the particular stimulus studied in this paper and its inherent symmetry properties, it was important to utilise a framework, such as neural fields, where these symmetry properties can be preserved. Another positive aspect of working with a relatively simple model is the small set of parameters that must be determined, in contrast to, for example, the study Chey et al. (1997) where a huge number of parameters must be determined heuristically. The strategy employed for setting model parameters took into account a number of important biological and mathematical constraints. One of the main ideas was to ensure that the model is operating close to its principal bifurcation where it will be most sensitive to subtleties of the stimulus input (Veltz 2011); the bifurcation analysis and other analytical results helped to ensure the model was operated in the right parameter regime. For the remaining parameters, we used numerical continuation to study the relationships between a given parameter and biologically relevant properties of the model’s solutions. We looked at these relationships over a wide range of parameter values and ensured that appropriate values were set. The general principles applied to tune parameters are applicable to a broad class of models that covers the various possible extensions proposed below. In existing studies of motion integration, behaviour was studied at fixed parameters (Chey et al. 1997; Simoncelli and Heeger 1998; Bayerl and Neumann 2004; Tlapale et al. 2010b); in certain cases the influence of a single term is tested by setting its weight to zero. Here the twoparameter continuation analysis allow us to determine an entire region of interest in parameter space. Further, the extended investigation in this region looked at the dynamical behaviour taking into account the effect of changing initial conditions. The twoparameter investigation also tells us that the behaviour produced is robust over entire regions of parameter space.
An important question in studies of motion perception is identifying exactly what happens in the first few hundred milliseconds of presentation. The most insightful work so far has come from ocular following experiments (Masson et al. 2000) and physiological recordings from individual neurons (Pack and Born 2001). In order to support the results in this paper we propose a psychophysics experiment that investigates the same stimulus described in this paper with incrementally increasing short presentation time in order to identify the qualitative change in percept distributions that were found in our model. We found a unimodal distribution in the shortterm dynamics and a trimodal distribution for the mediumterm dynamics. Furthermore, we predict that the D percept can only be seen as a transient and for long enough presentation times only the percepts H or V would be seen.
The model and results presented here not only captures a number of important aspects seen in experiment, but also forms a solid basis for further study of motion integration. One natural extension would be to lift the one dimensional approximation of the physical space. This would first of all allow for the validity of the original approximation to be tested, further, it would allow for other stimulus parameters such as the terminator ratio to be tested for further comparison with the results of Fisher and Zanker (2001). More complex aperture arrangements could also be considered such as the crossshaped stimulus studied in Castet and Zanker (1999). A directional bias that varied from subject to subject but generally towards the horizontal (Fisher and Zanker 2001) is a feature of experimental results that was not captured by our model. The most straightforward way to investigate this would be to consider an asymmetry in the model’s input that assumes stronger (or more numerous) inputs at the preferred direction. The model in Tlapale et al. (2010b), implemented in a similar framework to the one studied here, considers a much more detailed description taking into account filtering stages applied to the input along with feedforward and feedback interactions between MT and V1. The methods used in this paper would be applicable to models of much greater complexity though at the cost of increasing the intricacy of analytical computations and scale of numerical computations. The focus of this article was multistable perception in the first few seconds after stimulus onset, a similar model and stimulus could be used to study perceptual switches that are known to occur for extended presentations by considering an adaptation dynamic on a slow timescale such as the one studied in Curtu and Ermentrout (2004).
Notes
Acknowledgements
The authors would like to thank Guillaume Masson, Andrew Meso and Gregory Faye for their insight and for their helpful criticism of the work presented here. This research work received funding from CNRS, the European Community (through FACETS, ISTFET, Sixth Framework, No 025213, the ERC grant No 227747 (NERVI) and the Agence Nationale de la Recherche (ANR, NATSTATS)).
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