A recurrent network model of somatosensory parametric working memory in the prefrontal cortex

Abstract

A parametric working memory network stores the information of an analog stimulus in the form of persistent neural activity that is monotonically tuned to the stimulus. The family of persistent firing patterns with a continuous range of firing rates must all be realizable under exactly the same external conditions (during the delay when the transient stimulus is withdrawn). How this can be accomplished by neural mechanisms remains an unresolved question. Here we present a recurrent cortical network model of irregularly spiking neurons that was designed to simulate a somatosensory working memory experiment with behaving monkeys. Our model reproduces the observed positively and negatively monotonic persistent activity, and heterogeneous tuning curves of memory activity. We show that fine-tuning mathematically corresponds to a precise alignment of cusps in the bifurcation diagram of the network. Moreover, we show that the fine-tuned network can integrate stimulus inputs over several seconds. Assuming that such time integration occurs in neural populations downstream from a tonically persistent neural population, our model is able to account for the slow ramping-up and ramping-down behaviors of neurons observed in prefrontal cortex.

Figure 1

Schematic model architecture with asymmetric connectivity. Two mirror networks of positively and negatively monotonic neurons receive transient input respectively from positively and negatively tuned neurons in S2. Each network has an excitatory pyramidal cell population (squares) and an inhibitory interneuron population (circles). Neurons are divided into 12 groups per network. Synaptic connections are stronger within the same group than between two groups. The connectivity is asymmetrical, so that the activation threshold by stimulus is the lowest for neural group 1 and highest for neural group 12. Populations of inhibitory interneurons are shown as circles. The two networks interact through pyramid-to-interneuron connections, resulting in cross-inhibition. See text for more details.

Figure 2

Persistent neural activity of the parametric working memory model. (a) A positively monotonic, excitatory cell. Top panel: rastergrams, showing spikes in blocks of 10 trials, each block corresponding to a fixed stimulus frequency. The cell initially fires spikes at a few Hertz spontaneously. A transient stimulus (shaded) produces a large response, followed by persistent activity after the stimulus offset. The firing rate of both the transient response and persistent activity increases with the stimulus frequency. Middle panel: trial-averaged neural firing rate, where darker shades of gray represent increasing stimulus frequency. Bottom panel: the tuning curve shows the average rate in the last 5s of the delay period following each stimulus. (b) A negatively monotonic inhibitory interneuron, same plots as (a).

Figure 3

Sustained delay activity of prefrontal cortical neurons recorded from macaque monkeys during parametric working memory. (a) A positively monotonic neuron. (b) A negatively monotonic neuron. Same format as Figure 2

Figure 4

Diversity of tuning curves of persistent neural activity in prefrontal neurons and our model. (A) Examples of positively monotonic (left) and negatively monotonic (right) tuning curves from the experimental database. (B) Examples chosen to indicate the full variety of tuning curves from model simulations. Note the quasi-continuous nature of the curves, with small rate jumps. Filled circles: excitatory cells; open circles: interneurons.

Figure 5

Fine-tuning of parametric working memory model. Schematic illustration of a neural group with recurrent excitation. Left panel: network behavior as a function of the recurrent strength W_E→E and applied excitatory input, g_App. When W_E→E is above a critical value (e.g. point A), a bistability between a resting state and an active persistent state occurs in a range of g_App. This range shrinks to zero at the critical value of W_E→E, point B, which is called a ‘cusp’. Right panel: there is a trade-off between robust bistability but with a large gap in the firing rates of the two stable states (upper figure) and fine-tuning to the cusp where there can be a continuous range of firing rates (lower figure).

Figure 6

Bifurcation diagram of the finely tuned parametric working memory model, as a function of applied synaptic drive conductance, g_App. Synaptic drive is an offest conductance for demonstration purposes, that we add to the positively montonic neurons and subtract from the excitatory input to negatively monotonic neurons. A negative drive means the positively monotonic neurons have reduced synaptic excitation. A shift in any intrinsic neuronal parameter has a similar effect on the system. All the stable states are computed using the mean field theory for the entire network of twelve positively monotonic and twelve negatively monotonic, excitatory and inhibitory, neural groups. These persistent states are plotted as the firing rates of cells in neural group 3. (A) A quasi-continuum of stable firing rates is possible with correct tuning of applied synaptic drive. (B) An enlargement of the region near the quasi-continuous attractor indicates a discrete number of stable persistent states close to the number of neural groups, with small changes in firing rate between states. Portions of the curve with negative slopes are the branches of unstable states (dashed lines).

Figure 7

Time integration of a stimulus with different duration and amplitude. (A) The network can integrate a stimulus over a long time (1, 3 and 5 s), as shown by the rastergrams and population firing rates. (B) The firing rate of persistent activity (averaged between 3 and 6 s after the stimulus offset) is plotted as a function of stimulus duration, different curves correspond to different stimulus frequencies (4, 8, 12 and 16 Hz, with increasingly darker shades of gray). Note that linear dependence on the time duration of the stimulus occurs for moderate input strengths.

Figure 8

Diversity of delay period activity: tonic, early and late neurons. (A) Schematic diagram of an extended model with three neural populations (all are positively monotonic with the stimulus frequency). The first network (Tonic) shows tonic persistent activity and projects with strength g to a second network (Up), which integrates the inputs slowly to generate ramping-up activity during the delay. The third network (Down) displays a transient activation by the stimulus, and ramping-down time course of delay period activity due to the progressive inhibition from population 2. The trial-averaged firing rates for three different cells from each type of network are shown for 5 s following the stimulus frequency. (B) Neurons in population 2 ramp-up with a slope and a delay that depend on the input synaptic strength. Left panel: control (black, solid), and when the synaptic strength, g, from the tonic population 1 is reduced by one-third (gray, dashed). Right panel: when the time is scaled by half for the gray, dashed curve, the two time courses closely resemble each other.