Unraveling the sub-processes of selective attention: insights from dynamic modeling and continuous behavior

Abstract

Selective attention biases information processing toward stimuli that are relevant for achieving our goals. However, the nature of this bias is under debate: Does it solely rely on the amplification of goal-relevant information or is there a need for additional inhibitory processes that selectively suppress currently distracting information? Here, we explored the processes underlying selective attention with a dynamic, modeling-based approach that focuses on the continuous evolution of behavior over time. We present two dynamic neural field models incorporating the diverging theoretical assumptions. Simulations with both models showed that they make similar predictions with regard to response times but differ markedly with regard to their continuous behavior. Human data observed via mouse tracking as a continuous measure of performance revealed evidence for the model solely based on amplification but no indication of persisting selective distracter inhibition.

Appendix

Model dynamics

The basic formulation of a 1D neural field is given by the following equation

$$\tau \dot{u}\left( {x,t} \right) = - u\left( {x,t} \right) + h + s\left( {x,t} \right) + \int {w\left( {x - x'} \right)\, \cdot \,{\text{sig}}\left( {u\left( {x',t} \right)} \right){\text{d}}x'}$$

(1)

Here, u denotes the activation in the field, τ denotes the timescale, h denotes the resting level, s defines the input into the layer (as further defined below for each single layer) and w(x − x′) denotes the lateral interaction of units in the field. Notably, the term—u(x, t) leads to self-stabilizing dynamics around an attractor defined by the input, the resting level and the lateral interaction in the field.

Finally, sig denotes a sigmoid nonlinearity representing a soft threshold typical for neural population dynamics:

$${\text{sig}}\left( u \right) = \frac{1}{{1 + \exp \left( { - \beta \left( {u - \alpha } \right)} \right)}}$$

(2)

Here, β denotes the gain of the sigmoid function and α the soft threshold point.

Units in the field interact through a Gaussian activation kernel:

$$\begin{aligned}w\left( {x - x'} \right) &= w_{\text{gi}} + w_{\text{e}} \, \cdot \,\exp \left( {\frac{{ - \left( {x - x'} \right)^{2} }}{{2\sigma_{\text{e}}^{2} }}} \right) \\ &\quad - w_{\text{i}} \, \cdot \,\exp \left( {\frac{{ - \left( {x - x'} \right)^{2} }}{{2\sigma_{\text{i}}^{2} }}} \right) \end{aligned}$$

(3)

Here, w _gi denotes the weight of the global inhibitory interaction within the field, w _i denotes the weight of local inhibitory interaction within the field, w _e denotes the weight of local excitatory interaction in the field and σ denotes the spatial range of interaction in the field. Note that setting w _gi, w _i, w _e different to zero results in a mexican-hat-like interaction kernel.

The equation for the 2D association field follows Eq. (1) of the 1D field:

$$\tau \dot{u}\left( {x,y,t} \right) = - u\left( {x,y,t} \right) + h + s\left( {x,y,t} \right) + \iint {w\left( {x - x',y - y'} \right)\, \cdot \,{\text{sig}}\left( {u\left( {x',y',t} \right)} \right){\text{d}}x'{\text{d}}y'}$$

(4)

Notably, compared to the 1D interaction kernel (Eq. 3), the 2D interaction kernel is normalized by its’ square root. Since the current model did not use of the local inhibition term in the 2D layer, this term was omitted:

$$w\left( {x - x',y - y'} \right) = w_{\text{gi}} + w_{\text{e}} \, \cdot \,\sqrt {\exp \left( {\frac{{ - \left( {x - x'} \right)^{2} }}{{2\sigma^{2} }}} \right)\, \cdot \,\exp \left( {\frac{{ - \left( {y - y'} \right)^{2} }}{{2\sigma^{2} }}} \right)}$$

(5)

Parameters in the model were set by hand. To minimize the number of free parameters, most parameters were kept constant for all layers. Furthermore, parameters were kept equal for the two versions of the model. The parameters kept constant for all layers can be seen in Table 1. The parameters varying between layers can be seen in Table 2.

Table 1 Parameters kept constant for all layers of the model

Table 2 Parameters set individually for each layer

Model architecture

The model consisted of three layers in the activation-only version and four layers in the activation and inhibition version: the target layer, the association layer, the response layer and the additional distracter layer. All layers followed the activation dynamics as described by Eqs. 1–5. In the following, the inputs to each layer are described, laying down the connection architecture of the model.

The 1D target layer represented the information (here color) instructed as relevant. Parameters for the target layer can be seen in Table 2. The input to the target layer, representing the instruction of the currently relevant information, was set at the start of a trial. Importantly, the strength of self-excitation in the target layer led to self-stabilizing peaks, representing the memory of the once instructed information.

The input to the target layer was defined as

$$s_{\text{target}} \left( {x,t} \right) = wi_{\text{target}} \, \cdot \,I_{\text{target}} \left( {x,t} \right)$$

(6)

The input to the target layer consists of the currently presented colors I(x, t) of strength w _i.

The 2D association layer hosted the interaction between the 2D input (representing the stimuli on the screen) and the information from the target layer (the relevant information, i.e., color). In the inhibitory version, it also received inhibitory input from the distracter layer. The first dimension of the layer represented the color of the input, and the second dimension the magnitude of the presented numbers (for simplicity discretized in small and large). A peak at the location representing ‘red’ and ‘large’ hence represented a large red number, allowing a simple binding of stimulus features.

The target layer’s excitatory connection fed into the association layer as ridge over the color dimension. Hence, if the goal layer peaked at ‘red,’ this peak was fed into the association layer for ‘red’ across all possible magnitudes. Parameters for the association layer can be seen in Table 2. Similarly, the inhibition layer’s inhibitory connection also fed into the association layer as a valley over the color dimension. Hence, if the inhibition layer peaked at ‘blue,’ this peak inhibited the association layer for ‘blue’ across all possible magnitudes.

For the excitatory version, the input to the 2D association layer was defined as

$$\begin{aligned} s_{\text{association}} \left( {x,y,t} \right) & = wi_{\text{association}} \, \cdot \,I_{\text{association}} \left( {x,y,t} \right) \\ & \quad + \int {w_{\text{s}} \left( {x - x'} \right)\, \cdot \,{\text{sig}}\left( {u_{\text{target}} \left( {x',t} \right)} \right){\text{d}}x'} \\ \end{aligned}$$

(7)

For the inhibitory version, the input to the 2D association layer was extended by the inhibitory input from the distracter layer and hence defined as

$$\begin{aligned} s_{\text{association}} \left( {x,y,t} \right) & = wi_{\text{association}} \, \cdot \,I_{\text{association}} \left( {x,y,t} \right) \\ & \quad + \int {w_{\text{s}} \left( {x - x^{\prime } } \right) \, \cdot \,{\text{sig}}\left( {u_{\text{target}} \left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime}} \\ & \quad - \int {w_{\text{si}} \left( {x - x^{\prime}} \right)\, \cdot \,{\text{sig}}\left( {u_{\text{distracter}} \left( {x^{\prime},t} \right)} \right){\text{d}}x^{\prime}} \\ \end{aligned}$$

(8)

The input to the association layer consists of the currently presented colors and number of magnitudes I(x, y, t) weighted by w _I , the activation from the target layer convolved with a Gaussian weighting function (Eq. 3) w _s (with w _i = w _gi = 0, w _e = 3, σ = 3) and (in the inhibitory version) the activation from the distracter layer convolved with a Gaussian weighting function (Eq. 3) w _si (with w _i = 0.5, w _e = w _gi = 0, σ = 3).

The 1D response layer represented the response tendency of the model at any moment. It received its’ input from the association layer. From the peaks in the response layer, we calculated the movement tendencies of the system across time via a dynamic movement system (see next section). Parameters for the association layer can be seen in Table 2.

The input to the response layer was defined as

$$s_{\text{response}} \left( {x,t} \right) = \smallint w_{s} \left( {x - x'} \right)\, \cdot \,\int {{\text{sig}}\left( {u_{\text{association}} \left( {x,y,t} \right)} \right){\text{d}}y{\text{d}}x'}$$

(9)

The input to the response layer consists of the activation from the association layer convolved with a Gaussian weighting function (Eq. 3) w _s (with w _i = w _gi = 0 w _e = 3, σ = 2).

The 1D distracter layer hosted the interaction between shown stimuli and goal-relevant information, leading to a representation of information about material that was presented to the model, but was irrelevant with respect to the current goal. The distracter layer received excitatory input about the color of the shown stimuli and inhibitory input from the target layer. Similarly to the target layer, the strength of self-excitation in the inhibition layer led to self-stabilizing peaks, representing a memory of information that was distracting and should be ignored.

The input to the distracter layer was defined as

$$s_{\text{distracter}} \left( {x,t} \right) = wi_{\text{distracter}} \, \cdot \,I_{\text{distracter}} \left( {x,t} \right) - \int {w_{\text{si}} \left( {x - x'} \right)\, \cdot \,{\text{sig}}\left( {u_{\text{target}} \left( {x',t} \right)} \right){\text{d}}x'}$$

(10)

The input to the distracter layer consists of the currently presented colors I(x, t) convolved weighted with w _i, and the inhibition from the target layer convolved with a Gaussian weighting function (Eq. 3) w _si (with w _i = 10, w _e = w _gi = 0, σ = 3).

Calculation of movement trajectories

To read out the continuous movement tendency of the model on a one-dimensional virtual movement dimension, we used an attractor dynamics approach as it is well established in cognitive robotics (see Erlhagen and Bicho 2006).

The movement dimension corresponds to the dimensions of the response layer (here, 80 nodes), and the start position x _mov is set at the (neutral) middle of the dimension: x _mov = 40.

To update x _mov, we follow a two-step procedure for every time point in the simulation. This procedure constructs a dynamic system from the signal in the response layer. This dynamic system exhibits an attractor at the peak of activation, if present, in the layer.

In the first step, the position x _peak of the attractor is calculated by multiplying a ramp function along the x-axis with the activation peaks in the field, according to the formula

$$x_{\text{peak}} \left( t \right) = \frac{{\int {{\text{sig}}\left( {u\left( {x,t} \right)} \right)\, \cdot \,x{\text{d}}x} }}{{\int {{\text{sig}}\left( {u\left( {x,t} \right)} \right){\text{d}}x} }}$$

(11)

For a single peak of arbitrary width, x _peak locates the center of this peak. In the case of multiple peaks, x _peak locates the nearest location between these multiple peeks, indicating the best compromise for an indecisive movement.

In the second step, the tendency of movement of the systems is determined by constructing a one-dimensional dynamic system with one fix point at x _peak. The attractor is established by a Gaussian potential function similar to the Gaussian interaction kernel, with its’ peak at x _peak and a sigma of ½ of the size of the response layer (Fig. 4).

$$V\left( {x,t} \right) = - \frac{1}{{\sigma \sqrt {2\pi } }}\, \cdot \,\exp \left( { - \frac{1}{2}\, \cdot \,\left( {\frac{{x - x_{\text{peak}} \left( t \right)}}{\sigma }} \right)^{2} } \right)$$

(12)

Fig. 4

Calculation of simulated movement trajectories. Activation peaks in the response layer yield a Gaussian attractor in a dynamic movement system at the position of the peak x _peak. The current position on the movement dimension x _mov is updated by the change indicated by the dynamic system. When x _mov reaches a defined target area, the respective response is elicited

x _mov is then updated by the derivative

$$x_{\text{mov}} \left( t \right) = \frac{{{\text{d}}V\left( {x,t} \right)}}{{{\text{d}}x}}$$

(13)

Hence, it follows the movement tendency according to the potential function of Eq. 12 with a timescale of τ _mov = 10.

When x _mov reaches predefined target areas on the movement plane (left response: 0 < x < 23; right response: 57 < x < 80), a response is elicited and the trial ends.

Simulation of set-switching paradigm

We implemented a set-switching paradigm in which simulated participants responded to one of two stimuli that differed in their color and magnitude. Each trial started with the cueing of the currently task-relevant color by applying an input at one of three locations of the target layer (x = [10, 40, 70]), each of which represented one of the three colors used in our paradigm. With a delay of 12 cycles, two inputs were applied to the association layer representing the presentation of target and distracter stimuli. The peaks’ x- and y-locations in the association layer represented the stimuli’s properties in terms of color (x = [10, 40, 70]) and magnitude (y = [20, 60]). Target and distracter stimuli always differed in color and magnitude, that is, they always appeared at different locations on the x-axis and y-axis of the association layer. Simultaneously, with the presentation of target and distracter stimuli, two inputs were applied to the locations in the distracter layer representing both the current target and distracter color. As the distracter layer received additional inhibitory input from the target layer, only the distracter color survived in the layer, consequently inhibiting representations of task-irrelevant stimuli in the association layer. Responses were elicited when the movement parameter x _mov reached one of two predefined areas on the movement plane in the response layer (see above). Each response was followed by an intertrial interval of 80 cycles that allowed association and response layers to relax into their initial state.

We conducted two simulation studies: one with the full model in which both the target and the distracter layers influenced stimulus selection, and one with a reduced model in which the connections between distracter and association layers were severed, thus eliminating selective distracter suppression. In each simulation study, 30 simulated participants completed the set-switching experiment. Each experiment contained 16 switch trials in which target and distracter colors changed (8 trials following the PS- and 8 trials following the LI-logic). Each switch trial was followed by one repetition trial in which target and distracter colors remained constant (but stimulus magnitudes could change). The sequence of PS- and LI-switches was randomized across participants. In order to simulate interindividual variance due to differences in general speed of processing, we varied participants’ timescale parameters between-person by multiplying the values described above by a random factor ranging between 0.9 and 1.1.