Asymmetry in neural fields: a spatiotemporal encoding mechanism

Abstract

Neural field models have been successfully applied to model diverse brain mechanisms like visual attention, motor control, and memory. Most theoretical and modeling works have focused on the study of the dynamics of such systems under variations in neural connectivity, mainly symmetric connectivity among neurons. However, less attention has been given to the emerging properties of neuron populations when neural connectivity is asymmetric, although asymmetric activity propagation has been observed in cortical tissue. Here we explore the dynamics of neural fields with asymmetric connectivity and show, in the case of front propagation, that it can bias the population to follow a certain trajectory with higher activation. We find that asymmetry relates linearly to the input speed when the input is spatially localized, and this relation holds for different kernels and input shapes. To illustrate the behavior of asymmetric connectivity, we present an application: standard video sequences of human motion were encoded using the asymmetric neural field and compared to computer vision techniques. Overall, our results indicate that asymmetric neural fields are a competitive approach for spatiotemporal encoding with two main advantages: online classification and distributed operation.

Appendices

Appendix 1: Numerical simulations

The numerical simulation of Eqs. 3 and 16 was performed with a discrete version of Eq. 3,

$$\begin{aligned}&\frac{m(\theta ,t)}{\partial t} = \frac{1}{\tau } ( - m(\theta ,t) + \\&\qquad \left[\sum _{\theta ^{\prime }} w(\theta ^{\prime } \!-\! \theta ) m(\theta ^{\prime },t){\text{ d}}\theta ^{\prime } \!+\! C\left[1-\epsilon \!+\! \epsilon \cos (\theta -vt)\right] \!-\! T \right]^+ ), \end{aligned}$$

where we directly apply the 4th order Runge-Kutta method. To discretize the integral we use a trapezoidal rule. Here we use the parameters: ${\text{ d}}t=0.1, {\text{ d}}\theta =2 \pi /60, \tau =.15, J_0=-9.8, J_1=13.5, C=5, T=4.9$; and the total activity was computed over 1000 iterations. Numerically, the system is robust to simulate and we were able to obtain similar results using the Euler method.

In the case of 2D case the parameters were: $J_0=-.2, J_1=100, \sigma =21, {\text{ d}}r=1, {\text{ d}}\theta =2 \pi /\sigma$; and the total activity was computed over 100 iterations. Smaller kernel size and fewer iterations than in 1D were chosen to handle the dimension of input images ($160\times 120$ pixels). 1D and 2D implementations were performed in Matlab (Mathworks 2007) and parameters were selected by trial and error, except when indicated otherwise ($\sigma , {\text{ d}}r$).

Appendix 2: Neural field non-linearity

We translate the non-linearity outside the integral to simplify the analysis. Calling the input $I(x,t)=E(x,t) + h$ and writing it as $I(x,t)=\tau \frac{\partial \tilde{I}(x,t)}{\partial t} + \tilde{I}(x,t),$ we can rewrite Eq. 1 as,

$$\begin{aligned}&\tau \frac{\partial }{\partial t}\left(u(x,t) \!-\! \tilde{I}(x,t)\right) \!+\! u(x,t) - \tilde{I}(x,t) \nonumber \\&\quad \!=\! \int _{\Omega } w(|x^{\prime }- x|) f[u(x^{\prime },t)] {\text{ d}}x^{\prime }.\qquad \end{aligned}$$

(19)

Using the change of variable $u(x,t) - \tilde{I}(x,t) = \int _{\Omega } w(|x^{\prime }- x|) m(x^{\prime },t){\text{ d}}x^{\prime },$ and as w can be in general any function, we can express Eq. 19 as,

$$\begin{aligned}&\tau \frac{\partial m(x,t)}{\partial t} +m(x,t) = f\left[ u(x,t) \right], \\&\tau \frac{\partial m(x,t)}{\partial t} +m(x,t)\\&\quad =f\left[\int _{\Omega } w(x^{\prime }-x) m(x^{\prime },t){\text{ d}}x^{\prime }+\tilde{I}(x,t) \right]. \end{aligned}$$

Appendix 3: Input-asymmetry derivation

We take $\partial / \partial v$ of at both sides of Eq. 13 using $D=J^2_1 f^2_1 \cos (\Delta )^2 - 2 J_1 f_1 \cos (\Delta )\cos (\Delta +\beta )+1$ to obtain,

$$\begin{aligned} 0 =\frac{ \left( J_0 \frac{ \partial f_0}{ \partial \theta _c} \frac{\partial \theta _c}{\partial v} - \sin (\theta _c)\frac{\partial \theta _c}{\partial v} \right) \sqrt{D} -\frac{J_0 f_0 + \cos (\theta _c)}{2\sqrt{D}}\frac{\partial D}{\partial v}}{D}. \end{aligned}$$

(20)

As we minimize $\theta _c,$ we set $\partial \theta _c/ \partial v|_{v=v_m}=0$ in Eq. 20 and develop the term $\partial D / \partial v|_{v=v_m}=0$

$$\begin{aligned} \frac{ \partial D}{\partial v}\bigg |_{v=v_m}&= -J_1 f_1 \cos (\Delta ) \sin (\Delta ) +\sin (\Delta ) \cos (\Delta +\beta )\nonumber \\&+\cos (\Delta )\sin (\Delta +\beta )=0. \end{aligned}$$

(21)

A second order polynomia is obtained for $v_m$ from Eq. 21 recalling that $\Delta =\arctan (\tau v),$

$$\begin{aligned} v_m^2 \tau ^2 \sin (\beta )+ v_m \tau (J_1 f_1 - 2 \cos (\beta ))+\sin (\beta )=0. \end{aligned}$$