Weber's law in decision making: integrating behavioral data in humans with a neurophysiological model

Abstract

Recent single-cell studies in monkeys (Romo et al., 2004) show that the activity of neurons in the ventral premotor cortex covaries with the animal's decisions in a perceptual comparison task regarding the frequency of vibrotactile events. The firing rate response of these neurons was dependent only on the frequency differences between the two applied vibrations, the sign of that difference being the determining factor for correct task performance. We present a biophysically realistic neurodynamical model that can account for the most relevant characteristics of this decision-making-related neural activity. One of the nontrivial predictions of this model is that Weber's law will underlie the perceptual discrimination behavior. We confirmed this prediction in behavioral tests of vibrotactile discrimination in humans and propose a computational explanation of perceptual discrimination that accounts naturally for the emergence of Weber's law. We conclude that the neurodynamical mechanisms and computational principles underlying the decision-making processes in this perceptual discrimination task are consistent with a fluctuation-driven scenario in a multistable regime.

Figure 1.

Biophysical realistic computational model for a probabilistic decision-making network that performs the comparison of two mechanical vibrations applied sequentially (f1 and f2). The model implements a dynamical competition between different neurons. The network contains excitatory pyramidal cells and inhibitory interneurons. The neurons are fully connected (with synaptic strengths as specified in the text). Neurons are clustered into populations. There are two different types of population: excitatory and inhibitory. There are two subtypes of excitatory population, namely specific and nonselective. Specific populations encode the result of the comparison process in the two-interval vibrotactile discrimination task (i.e., whether f1 > f2 or f1 < f2). The recurrent arrows indicate recurrent connections between the different neurons in a population.

Figure 2.

Model results. A, Probability of correct discrimination as a function of the difference between the two presented vibrotactile frequencies to be compared (the horizontal dashed lines denote the thresholds of 15 and 85% correct classification). Error bars represent SEM. B, Weber's law for the vibrotactile discrimination task. The JND is calculated as one-half the difference between the frequency identified as higher than the standard on 85% of the trials and the frequency identified as smaller on 15% of the trials.

Figure 3.

Behavioral results. A, Mean probability of comparison higher responses (f1 < f2) as a function of the difference between the two presented vibrotactile frequencies to be compared. Error bars represent SEM. B, JND values for the different base frequencies, revealing Weber's law for the vibrotactile discrimination task. The JND is calculated as in the model (see Results).

Figure 4.

Minimal probabilistic decision-making neurodynamical network consisting of two self and mutually interacting neuronal populations. The activities of the specific populations encode the alternative choices. Continuous arrows represent excitatory connections between neurons in the same population with weight w₊. Dashed arrows represent inhibitory connections with weight w_I. External sensory input to the respective population is provided at rates λ and λ + Δλ.

Figure 5.

Bifurcation diagram of the minimal decision-making neural network as a function of the input λ. The curves plot the stable fixed points (“stationary attractor states”) of the reduced system defined by Equation 1 for population 1 (the results for population 2 are in the symmetric unbiased case identical). Black lines, Spontaneous state; gray lines, decision states. The vertical dashed lines delimit the multistable region.

Figure 6.

Computational principles underlying the different dynamical regimes shown in the bifurcation diagram of Figure 5: stable spontaneous state, multistable, and bistable.

Figure 7.

Decision-making behavior of the theoretical model predicted for the different dynamical regimes. The figure shows the critical discrimination Δλ value corresponding to an 85% correct performance level (difference threshold) as a function of the base frequency λ. Only in the region corresponding to the multistable regime did the difference threshold increase linearly as a function of the base frequency (i.e., consistent with Weber's law). The black line and the nonlinear gray curve shown in the graph correspond to a fit of the numerical results in the multistable and bistable regime, respectively. The vertical dashed lines delimit the multistable region according to the bifurcation diagram shown in Figure 5.