Test of a dynamic neural field model: spatial working memory is biased away from distractors

Abstract

Attention facilitates the encoding (e.g., Awh, Anllo-Vento, & Hillyard, J Cognit Neurosci 12(5), 840–847, 2000) and maintenance of locations in spatial working memory (Awh, Vogel, & Oh, Atten, Percept Psychophys 78(4), 1043–1063, 2006). When individuals shift their attention during the maintenance period of a spatial working memory task, their memory of a target location tends to be biased in the direction of the attentional shift (Johnson & Spencer, 2016). Dynamic field theory predicts that in certain conditions, inhibitory mechanisms will result in biases away from distractors presented during the maintenance period of the task. Specifically, dynamic field theory predicts that memory responses will be biased toward distractors that are near the target location and biased away from distractors that are farther from the target location. In two experiments, the current study tested adults in a spatial memory task that required memorization of a single target location. On a subset of trials, a distractor appeared during the memory delay at different distances and directions from the target location. In contrast to the prediction, memory responses were biased away from distractors that were near the target location and not biased by distractors that were far from the target location, providing challenges for, dynamic field theory and other theories of spatial working memory.

Appendix

The model used for these simulations was the same as the model used by Schutte and Spencer (2009) in Simulation Experiment 3 with the addition of a distractor input. This appendix presents the equations and parameters used for the model. For more details, see Schutte and Spencer (2009).

Activation in the perceptual layer (PF) is governed by the equation:

$$\tau \dot {u}(x,t)\;=\; - u(x,t)+{h_u}+\int {{c_{uu}}} (x - {x^\prime }){\Lambda _{uu}}(u({x^\prime },t)){\text{d}}{x^\prime } + S_{ref}(x,t) + S_{tar}(x,t) + S_{distractor}(x,t) - \int {{c_{uv}}} (x - {x^\prime }){\Lambda _{uv}}(u({x^\prime },t)){\text{d}}{x^\prime } + q\mathop \int \nolimits^{} dx^{'} g_{{noise}} \left( {x - x^{'} } \right)\varepsilon \left( {x^{'} ,t} \right),$$

where the rate of change of the activation level for each neuron across the spatial dimension, x, as a function of time, t, is determined by the current activation level, u(x,t), the resting level, h_u, self-excitatory projections, $\int {{c_{uu}}(x - {x^\prime })} {\Lambda _{uu}}(u({x^\prime },\;t))d{x^\prime }$ inhibitory projections from the Inhibitory layer (Inhib), $\int {{c_{uv}}(x - {x^\prime })} {\Lambda _{uv}}(u({x^\prime },\;t))d{x^\prime }$, reference input, S_ref(x,t), target input, S_tar(x,t), distractor input, S_distractor(x,t), and spatially correlated noise, $q\int {{\text{d}}{x^\prime }} {g_{{\text{noise}}}}(x - {x^\prime })\xi ({x^\prime },\;t)$. The reference, target, and distractor inputs are Gaussians (see Table 1 for the width and strength of each Gaussian). The excitatory and inhibitory projections are determined by the convolution of a Gaussian kernel with a sigmoidal threshold function. The Gaussian kernel is specified by:

$$c(x - {x^\prime })\;=\;c\;\exp \left[ { - \frac{{{{(x - {x^\prime })}^2}}}{{2{\sigma ^2}}}} \right]-k,$$

with strength, c, width, σ, and resting level, k. The level of activation required to enter into the interaction is determined by the following sigmoid function:

$$\Lambda (u)\;=\;\frac{1}{{1+\exp [ - \beta u]}},$$

where β is the slope of the sigmoid. The slope determines whether neurons close to threshold (i.e., 0) contribute to the activation dynamics with lower slope values permitting graded activation near threshold to influence performance, and higher slope values ensuring that only above-threshold activation contributes to the activation dynamics.

Table 1 Parameter values for simulations

The inhibitory layer, inhib, is governed by a similar equation:

$$\begin{gathered} \tau \dot {v}\left( {x,t} \right)=~ - v\left( {x,t} \right)+{h_v}+~\smallint {c_{vu}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{vu}}\left( {u\left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime} - ~\smallint {c_{vw}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{vw}}\left( {w\left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime} \hfill \\ +~q\smallint {\text{d}}x^{\prime}{g_{{\text{noise}}}}\left( {x - x^{\prime}} \right)\varepsilon \left( {x^{\prime},t} \right), \hfill \\ \end{gathered}$$

As in the equation for PF, $\dot {v}(x,t)$ is the rate of change of the activation level for each neuron across the spatial dimension x, as a function of time, t, $v(x,t)$ is the current activation in the field, and, h_v, sets the resting level of the field. Inhib (v) receives input from both PF(u), $\int {{c_{vu}}} (x - {x^\prime }){\Lambda _{vu}}(u({x^\prime },t)){\text{d}}{x^\prime }$, and SWM(w), $\int {{c_{vw}}} (x - {x^\prime }){\Lambda _{vw}}(w({x^\prime },t)){\text{d}}{x^\prime }$. These projections are defined by the convolution of a Gaussian kernel with a sigmoidal threshold function using the same equations as the interaction in PF(u). As in PF(u), the final input to the field is noise, $q\int d {x^\prime }{g_{noise}}(x - {x^\prime })\zeta ({x^\prime } - t)$.

The SWM layer (w) is governed by a similar equation:

$$\tau \dot {w}\left( {x,t} \right)=~ - w\left( {x,t} \right)+{h_w}+~\int {c_{ww}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{ww}}\left( {w\left( {x^{\prime},t} \right)} \right)dx^{\prime} - ~\int {c_{wv}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{wv}}\left( {v\left( {x^{\prime},t} \right)} \right)dx^{\prime}+~\int {c_{wu}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{wu}}\left( {u\left( {x^{\prime},t} \right)} \right)dx^{\prime}+{c_s}{S_{ref}}\left( {x,t} \right)+{c_s}{S_{tar}}\left( {x,t} \right)+~+~q\int dx^{\prime}{g_{noise}}\left( {x - x^{\prime}} \right)\varepsilon \left( {x^{\prime},t} \right),$$

As in the previous equations, w(x,t) is the current activation in the field, and h_w is the resting level. Inputs to SWM include self-excitation, $\smallint {c_{ww}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{ww}}\left( {w\left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime}$, lateral inhibition from Inhib, $\smallint {c_{wv}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{wv}}\left( {v\left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime}$, and input from PF, $\smallint {c_{wu}}\left( {x - x^{\prime}} \right){{{\varvec{\Lambda}}}_{wu}}\left( {u\left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime}$. SWM also receives weak direct reference input, S_ref(x,t),and target input, S_tar(x,t), all scaled by c_s. The final input to the field is spatially correlated noise, $\smallint dx^{\prime}{g_{noise}}\left( {x - x^{\prime}} \right)\varepsilon \left( {x^{\prime},t} \right)$.

With the exception of the addition of a distractor input, parameter values were the same as in Schutte and Spencer (2009; see Table 1). The size of the fields was 397 units with 1.2 units equal to 1 degree, and noise strength was set to 0.135 with noise width, the spatial spread of noise, set to 1.