‘Two vs one’ rivalry by the Loxley–Robinson model

Abstract

We apply the competitive model of Loxley and Robinson (Phys Rev Lett 102:258701, 2009. doi:10.1103/PhysRevLett.102.258701) to the study of a special case of visual rivalry. Three-peaked inputs with maxima at symmetrical locations are introduced, and the role of three-bump configurations is then considered. The model yields conditions for what can be interpreted as a bistable percept analogous to the one-dimensional version of a competition between the central and flanking parts of an image.

A Appendix: stability criterion

Stability is often studied by application of the Evans technique (see [10,11,12, 26] and refs therein), assuming that time-dependent deviations from the stationary solutions $u_s(x)$, $\theta _s(x)$ have the form

$$\begin{aligned} \left\{ \begin{array}{lll} u(x, t)&{}=&{}\displaystyle u_s(x)+\delta u(x) \, \hbox {e}^{\lambda t} \\ \theta (x, t)&{}=&{}\displaystyle \theta _s(x)+\delta \theta (x) \, \hbox {e}^{\lambda t} . \end{array} \right. \end{aligned}$$

(12)

where $\delta u$, $\delta \theta$ are regarded as small magnitudes. Inserting (3), (12) into (1), one can expand the resulting expressions to first order in $\delta u$, $\delta \theta$. Such an expansion is performed with the help of the formal rule $\displaystyle {\delta \over \delta g(z)}\Theta \left( g(z) \right) = \delta _\mathrm{D}( g(z) )= {1 \over |g'(z)|} \delta _\mathrm{D}(z),$ valid inside integrals, where $\delta _\mathrm{D}$ denotes Dirac’s delta. This procedure enables us to factor out the time dependence and obtain a set of $4N_s$ equalities no longer t-dependent, just x-dependent. Next, we consider as specific x values the boundary points themselves

$$\begin{aligned} \left( x_1, x_2, \dots , x_{2N_s-1}, x_{2N_s} \right)= & {} \left( a_1-{m_1 \over 2}, a_1+{m_1 \over 2}, \dots ,\right. \nonumber \\&\left. a_{N_s} -{m_{N_s} \over 2}, a_{N_s}+{m_{N_s} \over 2} \right) .\nonumber \\ \end{aligned}$$

(13)

For the particular values $x= x_i$, $i=1,\dots , 2N_s$, those general equalities reduce to a set of equations which can be arranged into the form

$$\begin{aligned} \mathcal {M} \left( \begin{array}{c} \delta \vec {u} \\ \delta \vec {\theta } \end{array} \right) =\left( \begin{array}{c@{\quad }c} \widehat{\mathcal {D}}_f(\lambda ) \, I_{2N_s}-\mathcal {A}&{}\mathcal {A}\\ -\kappa I_{2N_s} &{} \widehat{\mathcal {D}}_s(\lambda ) \, I_{2N_s} \end{array} \right) \left( \begin{array}{c} \delta \vec {u} \\ \delta \vec {\theta } \end{array} \right) =0\nonumber \\ \end{aligned}$$

(14)

with

$$\begin{aligned} \begin{array}{lll} \delta \vec {u}= \left( \begin{array}{c} \delta u(x_1) \\ \vdots \\ \delta u(x_{2Ns}) \end{array} \right) &{} &{} \delta \vec {\theta }= \left( \begin{array}{c} \delta \theta (x_1) \\ \vdots \\ \delta \theta (x_{2Ns}) \end{array} \right) , \end{array} \end{aligned}$$

(15)

$\widehat{\mathcal {D}}_f(\lambda )\equiv \tau _f\lambda +1$, $\widehat{\mathcal {D}}_s(\lambda )\equiv \tau _s\lambda +1$. $I_{2N_s}$ is the $2N_s \times 2N_s$ identity matrix and $\mathcal {A}$ is a $2N_s \times 2N_s$ matrix with coefficients

$$\begin{aligned} \mathcal {A}_{ij}= {w(x_i-x_j)\over |u_s'(x_j)-\theta _s'(x_j) |} ,\quad 1 \le i,j \le 2 N_s, \end{aligned}$$

(16)

where the $x_i$’s and $x_j$’s are the region limits in the notation (13). The denominators are easily obtained from (3) which, after differentiation w.r.t. the only present argument, supply

$$\begin{aligned} \begin{array}{lll} u_s'(x_j)-\theta _s'(x_j)&{}=&{} (1-\kappa ) \, u_s'(x_j) \\ &{}=&{}\displaystyle (1-\kappa ) \, \sum _{r=1}^{N_s} \Big [ w\left( x_j-a_r+{m_r\over 2} \right) \\ &{}&{} -w\left( x_j-a_r-{m_r \over 2} \right) +h'(x_j) \Big ]. \end{array} \end{aligned}$$

(17)

Existence of nontrivial solutions to system (14) requires $\det (\mathcal {M})= 0$. (This is often—perhaps not quite correctly—called ‘eigenvalue equation’ because, like in a true eigenvalue problem, the solutions are the different values of a scalar parameter which cause a determinant to vanish.) Further, by virtue of the block structure of $\mathcal {M}$, its determinant amounts to

$$\begin{aligned} \det (\mathcal {M})= \det \left[ \widehat{\mathcal {D}}_f(\lambda ) \widehat{\mathcal {D}}_s(\lambda ) I_{2N_s} -(\widehat{\mathcal {D}}_s(\lambda )-\kappa ) \mathcal {A} \right] . \end{aligned}$$

(18)

Moreover, for $\widehat{\mathcal {D}}_s(\lambda )-\kappa \ne 0$, it is possible to divide by $\widehat{\mathcal {D}}_s(\lambda )-\kappa$ and, doing as the authors of Ref. [26] did, we can just study the zeros of the following Evans function

$$\begin{aligned} \mathcal {E}(\lambda )= \det \left( {\widehat{\mathcal {D}}_f(\lambda ) \widehat{\mathcal {D}}_s(\lambda )\over \widehat{\mathcal {D}}_s(\lambda )-\kappa } I_{2N_s} -\mathcal {A} \right) . \end{aligned}$$

(19)

In the ‘Evans sense’, stability of the $u_s$, $\theta _s$ solutions requires that all the $\lambda$’s satisfying $\mathcal {E}(\lambda )= 0$ have Re$(\lambda ) < 0$. This criterion is easy to understand by recalling the form of the Ansatz (12) and realizing how the character of $\lambda$ determines the qualitative nature of the deviations of u and $\theta$ from $u_s$ and $\theta _s$, respectively.

Note that by (19) and relations (16), (17), the Evans function $\mathcal {E}(\lambda )$ depends, through the presence of the $\mathcal {A}$ matrix, on the form of the particular h(x), $h'(x)$. Furthermore, because this matrix also depends directly on the found centres and widths—the $a_i$, $m_i$, $1 \le i \le N_i$—the shape of every possible Evans function is determined by these particular values as well.