# (p.301) Appendix A Some mathematical aspects of masking models

# (p.301) Appendix A Some mathematical aspects of masking models

(p.301) Appendix A

Some mathematical aspects of masking models

# A.1. Multiplicative (shunting) and additive equations

In Chapter 5, we introduced two ‘generic’ formalisms that have been used extensively in neural modeling (e.g. Grossberg 1988; Koch and Segev 1989). The first equation has the form of Hodgkin–Huxley equation and is written as

*g*

_{d}and

*g*

_{h}represent variable (active) conductances through which input signals modulate the membrane potential

*V*

_{m}. In this sense, this is an ‘active’ model of a neuron. Because of the multiplicative interactions between the input and the membrane potential, the model is also known as the multiplicative or shunting model (e.g. Grossberg 1988). A simpler version of this model is the ‘additive’ model

The BCS model (Chapter 4, section 4.7) and the RECOD model (Chapter 5) build their dynamic network representations using equations (A1) and (A2). We will show below that the equations used in Bridgeman’s and Weisstein’s models are of the additive type.

(p.302) Consider first the equations of Bridgeman’s Hartline–Ratliff inhibitory network:

*r*) is the firing rate of neuron

_{i}(t*i*at time

*t, e*) is the excitatory input to the

_{i}(t*i*th neuron,

*w*is the synaptic weight for the connection from the

_{j,i}*j*th neuron to the

*i*th neuron,

*r*

_{j,i}^{0}is the firing threshold for the connection between the

*j*th and the

*i*th neuron, and

*n*is the number of neurons in the network. Equation (A3) is a

**difference equation**and equation (A2) is a

**differential equation**. The two equations can be compared by approximating the derivative by a backward-difference formula. Following the transformations used by Grossberg (1988), let

*A*= 1 and with

*e*) and${\sum}_{j=1}^{n}{w}_{j,i}\left[{r}_{j}(t-|i-j|)-{r}_{j,i}^{0}\right]$corresponding to the excitatory and inhibitory inputs, respectively. Note that while the state variables in the two equations correspond to different physiological variables (membrane potential in equation (A3) and firing rate in equation (A5)), the Hartline–Ratliff model applies thresholds directly to firing rates, while network formulations of equations (A1) and (A2) apply thresholds to membrane potentials.

_{i}(tAs discussed in Chapter 4, the building blocks of Weisstein’s model are the two-factor Rashevsky–Landahl equations

*ε*and

_{j}*j*are the excitatory and inhibitory factors, respectively, to the

_{j}*j*th neuron. These factors can be interpreted and modeled as excitatory and inhibitory neurotransmitters. Alternatively, we can introduce an additional ‘interneuron’, as shown in Figure A1, to express this model as a small circuit of two neurons (Öğmen 1993, Appendix B).

Let the two neurons in Figure A1 obey the additive equations

*x*

_{1}given by

*–*

**α**= α_{j},**β**= A_{j}*B*=

_{j},**γ***a*, and

_{j}– b_{j},**δ**= b_{j},**η**= B_{f}**Γ**=

*h*.

_{j}# (p.304) A.2. The approach of Anbar and Anbar

Anbar and Anbar (1982) proposed a mathematical model for masking based on three assumptions: step visual response function (VRF) and exponential decay, temporal integration, and lateral inhibition.

## A.2.1. Step visual response function

The VRF *v(t*) for an input of intensity *I* applied at time *t* = 0 and of duration *t* _{0} is assumed to have the following form:

*α = kI*where k and γ represent two constants. Thus the response is assumed to increase stepwise to its level while the stimulus is on and to decay exponentially after the stimulus is turned off.

^{γ}Using equations (A1) and (A2) we also obtain an exponential rise and decay in the response. Anbar and Anbar use step increase as a simplification and assume specific power relations for response magnitude and decay rate.

## A.2.2. Temporal integration

Temporal integration is used as a linking assumption for perceived brightness *V(I*):

## A.2.3. Lateral inhibition

Finally, Anbar and Anbar assume that masking results from a type of simultaneous brightness contrast effected by lateral inhibition such that the weaker of the two stimuli is suppressed by the stronger. This suppression is assumed to be a step decrease by an amount proportional to the *p*th power of the ratio between the two VRFs at the onset of the masking stimulus. For example, assume that the target and
(p.305)
the mask have intensities *I* _{T} and *I* _{M}, respectively, with *I* _{T} < *I* _{M}. Assume also that the mask is applied while the target is on. By equation (A13), when the target is on, the VRF for the target is *cI* _{T} ^{β}. Similarly, the VRF for the mask is *cI* _{M} ^{β}. Since the mask is stronger than the target, the VRF for the target will be suppressed by the ratio

*t*=

*τ t*

_{0}, the VRF will be suppressed by the ratio

Inspection of these expressions shows how a U-shaped backward masking function is obtained. First, consider equation (A17). For a fixed mask intensity, larger values of τ lead to smaller values of this ratio. Since VRF is multiplied by this ratio, smaller values of the ratio imply larger suppression of activity. Thus, from the perspective of the ratio (A17), masking becomes more effective as ISI increases. On the other hand, inspection of (A18) shows that the actual suppression of VRF is given by the product of this ratio and an exponentially decaying VRF. Because larger values of ISI correspond to smaller values of the exponentially decaying VRF, from the perspective of the ongoing VRF, masking becomes less and less effective as ISI increases. Putting the two opposing tendencies together, we obtain a U-shaped function. It should be noted that a secondary effect of the drop in activity is the concomitant change in the rate of subsequent decay. However, Francis (2000) showed that a U-shaped function can be obtained without the change in decay rate, although joint changes in amplitude and decay rate produce a U-shaped masking function that is quantitatively different from that obtained solely by a change in amplitude.

# (p.306) A.3. Concepts of mask and target blocking

The simple mathematical formulation of the Anbar and Anbar model leads to an important insight on how a U-shaped masking function can be produced. As mentioned above, because ratio (A15) depends on the strength of the target, masking is more efficient when the mask is delayed. However, because the activity is decaying, the mask has to be applied while the activity remains at a significant level; otherwise, most of the target’s activity remains intact, leading to a high level of visibility. These two constraints together lead to an ‘optimal’ intermediate ISI wherein maximum masking occurs. Francis (2000) formulated this intuition into a general property that he called **mask blocking**. Mask blocking is defined as the blocking of the mask’s inhibitory effect by the target’s activity (cf. numerator in equation (A15)). The complementary concept, **target blocking**, occurs when the mask blocks the target signal from contributing to the visibility of the target. Assume, for example, that the inhibitory signal of the mask is largely independent of the activity of the target. In this case the inhibitory activity of the mask will reduce the activity of the target without a reciprocal interference from the target and thus target blocking will occur. More detailed analysis and predictions can be found in Francis and Cho (2005).