From fixed points to chaos: three models of delayed discrimination

Abstract

Working memory is a crucial component of most cognitive tasks. Its neuronal mechanisms are still unclear despite intensive experimental and theoretical explorations. Most theoretical models of working memory assume both time-invariant neural representations and precise connectivity schemes based on the tuning properties of network neurons. A different, more recent class of models assumes randomly connected neurons that have no tuning to any particular task, and bases task performance purely on adjustment of network readout. Intermediate between these schemes are networks that start out random but are trained by a learning scheme. Experimental studies of a delayed vibrotactile discrimination task indicate that some of the neurons in prefrontal cortex are persistently tuned to the frequency of a remembered stimulus, but the majority exhibit more complex relationships to the stimulus that vary considerably across time. We compare three models, ranging from a highly organized line attractor model to a randomly connected network with chaotic activity, with data recorded during this task. The random network does a surprisingly good job of both performing the task and matching certain aspects of the data. The intermediate model, in which an initially random network is partially trained to perform the working memory task by tuning its recurrent and readout connections, provides a better description, although none of the models matches all features of the data. Our results suggest that prefrontal networks may begin in a random state relative to the task and initially rely on modified readout for task performance. With further training, however, more tuned neurons with less time-varying responses should emerge as the networks become more structured.

Figure 1

The experimental task and sample neurons. A Task protocol: a mechanical probe is lowered (PD) and then the monkey grasps a key (KD) to signal readiness. Following a delay of 1.5–3 s, the first stimulus is delivered followed by a 3 s delay and the second stimulus. The monkey then releases the key (KU) and presses one of two buttons (PB) to report whether f₁>f₂ or f₁<f₂. B Performance: percent correct on each of the 10 stimulus pairs used. C–F PSTHs of four neurons during the task. Shaded areas denote the stimulus periods, and color indicates the frequency of the first stimulus according to the colorbars. C A positively tuned neuron. D A negatively tuned neuron. E A neuron that is negatively tuned during the first stimulus and positively tuned during the delay. F A neuron with strong temporal modulation, but no tuning to the frequency of the stimulus.

Figure 2. The three models

Columns 1 and 2 illustrate the models we consider, column 3 shows the output of each model, and column 4 demonstrates model performance. Stimulus presentations times are indicated in gray in column 3, and the decision time is at the end of the time period shown. Red and blue traces correspond to f₁>f₂ and f₁<f₂, respectively, as indicated by the schematics above. Correct responses occur when the blue traces are positive and the red traces are negative at decision time. In column 4, performance is shown as the fraction of “f₁>f₂” responses for the entire f₁-f₂ plane (see colorbar above), with the 10 experimental frequency pairs indicated by the green circles. The LA model is composed of populations of spiking neurons (1^st column) receiving positive and negative tuned input during the stimulus (shaded) with self-excitation and mutual inhibition. The output is defined as the difference between the firing rates of the positive and negative populations. The RN model is composed of 1500 rate units randomly connected to each other, creating chaotic dynamics. 30% of the neurons receive external input during the stimuli, and the output is a trained linear readout from the entire network. The TRAIN model starts from a similar set up as the RN network, but all connections are trained (depicted in red) using the Hessian-Free algorithm. The output is a linear readout as for the RN model.

Figure 3. Activity of sample neurons from the models

The firing rates of two neurons taken from each model. The color represents the frequency of the first stimulus, from 10 Hz (blue) to 34Hz (red). Gray shadings denote the stimulus presentation periods.

Figure 4

Consistency of frequency tuning. Firing rates were fit according to r = a₀ + a₁ f₁. Values of a₁ at the end of the delay were compared to those during the stimulus (1) or mid-delay (2). 3 Correlation of a₁ values across the population using either the stimulus (blue) or mid-delay (green) as a reference.

Figure 5

Linear extraction by a modified PCA (Machens et al., 2010) of a time invariant signal that most strongly reflects the coding of stimulus frequency during the delay period. The extraction is done on half the trials, and the projection on the other half. Note that even the random network has a roughly time-invariant component.

Figure 6

Temporal evolution of the fraction of significantly tuned neurons in the data and models. The models were simulated with an identical number of trials as the data to allow evaluation of significance. Note that only the data show an increase in the number of significantly tuned neurons during the delay period.