Modelling the differential effects of prisms on perception and action in neglect

Abstract

Damage to the right parietal cortex often leads to a syndrome known as unilateral neglect in which the patient fails to attend or respond to stimuli in left space. Recent work attempting to rehabilitate the disorder has made use of rightward-shifting prisms that displace visual input further rightward. After a brief period of adaptation to prisms, many of the symptoms of neglect show improvements that can last for hours or longer, depending on the adaptation procedure. Recent work has shown, however, that differential effects of prisms can be observed on actions (which are typically improved) and perceptual biases (which often remain unchanged). Here, we present a computational model capable of explaining some basic symptoms of neglect (line bisection behaviour), the effects of prism adaptation in both healthy controls and neglect patients and the observed dissociation between action and perception following prisms. The results of our simulations support recent contentions that prisms primarily influence behaviours normally thought to be controlled by the dorsal stream.

Appendix: Neural engineering framework

The NEF was developed by Eliasmith and Anderson (2003) and specifies how complex networks of neurally plausible neurons can be constructed. In this model, we use a spiking leaky integrate and fire (LIF) neuron model for all neurons. The form of this model is:

$$a_{i} \left( x \right) = G_{i} \left[ {\alpha_{i} e_{i} x + J_{i}^{\text{bias}} } \right],$$

(12)

where G _i is the spiking LIF nonlinearity governed by:

$$\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{1}{{\tau^{\text{RC}} }}\left( {J\left( t \right)R_{\text{m}} - V\left( t \right)} \right),$$

(13)

such that the neuron spikes when the voltage V passes a threshold. In these equations, a _i (x) is the activity of neuron i in response to the input, x is the state-space input to the neuron, $\mathcal{T}^{\rm RC}$ is the membrane time constant, α is the input gain, e _i is the encoding vector that relates neural activity to the state space, J ^bias is a background current, and R _m is the membrane resistance. The LIF model is also governed by $\mathcal{T}^{\rm ref}$, an absolute refractory period applied after each spike.

The α and J ^bias parameters jointly specify the range of inputs each neuron is sensitive to, as well as the maximum firing rate of the neuron. All neurons in this model were randomly selected to have maximum firing rates between 200 and 400 Hz, with an even distribution. These are high for cortical firing. However, lower maximum firing rate values can be used with the same performance, although the number of neurons must increase proportionally. This comes at high computational cost, hence significantly increasing simulation times. To avoid undue simulation time, we left the firing rates high. In addition, J ^bias values were randomly selected to give neurons with x intercepts evenly distributed over the range of represented values for a given population. The parameters $\mathcal{T}^{\rm ref}$ and $\mathcal{T}^{\rm RC}$ were set to the biologically plausible values of 0.001 and 0.02 ms, respectively.

Equations (12) and (13) determine how a state space, x, is encoded into neural spikes. The first principle of the NEF suggests that the information carried by such a spike train can be linearly decoded. That is,

$$\hat{x} = \sum \limits_{i = 1}^{N} a_{i} \left( x \right)d_{i}$$

(14)

where $\hat{x}$ is the decoded estimate of the state space, N is the number of neurons in the population, a _i (x) is the activity of neuron i given the input (defined by 13), x is the input to each neuron in the population, d _i is the decoding weight of neuron i. The decoders are found by solving for the optimal least squares linear decoders. That is, minimizing

$$E = \int \nolimits \left[ {x - \hat{x}} \right]^{2} {\text{d}}x,$$

(15)

with respect to d _i .

Principle 2 of the NEF suggests that a population decoding can be made to decode arbitrary functions by finding appropriate decoding weights. That is, we can substitute an arbitrary f(x) for $\hat{x}$ in (15) and perform the same minimization to find decoders, d ^f(x) _i , that can be used to estimate the function f(x).

Using these decoders, the NEF suggests a method for building many layer (and recurrent) networks. For instance, if we define the encoding of a second population, b, analogously to (12), we have

$$b_{j} \left( y \right) = G_{j} \left[ {\alpha_{j} e_{j} y + J_{j}^{\text{bias}} } \right].$$

(16)

If we wish to compute the simple transformation y = x from population a to population b, we can substitute $y = x \approx \hat{x}$ from (14) into (16) to determine connection weights:

$$\begin{aligned} b_{j} \left( y \right) & = G_{j} \left[ {\alpha_{j}e_j \sum \limits_{i = 1}^{N} a_{i} \left( x \right)d_{i} + J_{j}^{\text{bias}} } \right] \\ b_{j} \left( y \right) & = G_{j} \left[ {\mathop \sum \limits_{i = 1}^{N} w_{ij} a_{i} \left( x \right) + J_{j}^{\text{bias}} } \right] \\ \end{aligned}$$

where w _ij  = α _j e _j d _i .

Using this method, neuronal populations can be made to compute a wide variety of functions. Those relevant for this model are described in the ‘Component Descriptions’ section of the ‘Methods’. Neuronal populations can then be chained together through synaptic connections to construct large, complex, neurally plausible networks. See Eliasmith and Anderson (2003) for a more rigorous derivation and further details of the NEF.