Computational aspects of feedback in neural circuits

Abstract

It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant real-time computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints. Although this article examines primarily the computational role of feedback in circuits of neurons, the mathematical principles on which its analysis is based apply to a variety of dynamical systems. Hence they may also throw new light on the computational role of feedback in other complex biological dynamical systems, such as, for example, genetic regulatory networks.

Figure 1. Computational Architectures Considered in Theorems 1 and 2

(A) A fixed circuit C whose dynamics is described by the system (Equation 3). (B) An arbitrary given n^th order dynamical system (Equation 4) with external input u(t). (C) If the input v(t) to circuit C is replaced by a suitable feedback K(x(t),u(t)), then this fixed circuit C can simulate the dynamic response z(t) of the arbitrarily given system shown in B, for any input stream u(t). (D) Arbitrary given FSM A with l state. (E) A noisy fading-memory system with feedback can reliably reproduce the current state A(t) of the given FSM A, except for timepoints t shortly after A has switched its state.

Figure 2. State-Dependent Real-Time Processing of Four Independent Input Streams in a Generic Cortical Microcircuit Model

(A) Four input streams, each consisting of eight spike trains generated by Poisson processes with randomly varying rates r_i(t),i = 1,…4, rates plotted in (B); all rates are given in Hz. The four input streams and the feedback were injected into disjoint sets of neurons in the circuit. (C) Resulting firing activity of 100 out of the 600 I&F neurons in the circuit. Spikes from inhibitory neurons marked in red. (D) Target activation times of the high-dimensional attractor (blue shading), spike trains of two of the eight I&F neurons that were trained to create the high-dimensional attractor by sending their output spike trains back into the circuit, and average firing rate of all eight neurons (lower trace). (E,F) Performance of linear readouts that were trained to switch their real-time computation task depending on the current state of the high-dimensional attractor: output 2 · r ₃(t) instead of r ₃(t) if the high-dimensional attractor is on (E), output r ₃(t) + r ₄(t) instead of | r ₃(t) – r ₄(t) | if the high-dimensional attractor is on (F). (G) Performance of linear readout that was trained to output r ₃(t) · r ₄(t), showing that another linear readout from the same circuit can simultaneously carry out nonlinear computations that are invariant to the current state of the high-dimensional attractor.

Figure 3. Representation of Time for Behaviorally Relevant Timespans in a Generic Cortical Microcircuit Model

(A) Afferent circuit input, consisting of a cue in one channel (red) and random spikes (freshly drawn for each trial) in the other channels. (B) Response of 100 neurons from the same circuit as in Figure 2, which has here two coexisting high-dimensional attractors. The autonomously generated periodic bursts with a periodic frequency of about 8 Hz are not related to the task, and readouts were trained to become invariant to them. (C,D) Feedback from two linear readouts that were simultaneously trained to create and control two high-dimensional attractors. One of them was trained to decay in 400 ms (C), and the other in 600 ms (D) (scale in nA is the average current injected by feedback into a randomly chosen subset of neurons in the circuit). (E) Response of the same neurons as in (B), for the same circuit input, but with feedback from a different linear readout that was trained to create a high-dimensional attractor that increases its activity and reaches a plateau of 600 ms after the occurrence of the cue in the input stream. (F) Feedback from the linear readout that creates this continuous high-dimensional attractor.

Figure 4. A Model for Analog Real-Time Computation on External and Internal Variables in a Generic Cortical Microcircuit (Consisting of 600 Conductance-Based HH Neurons)

(A,B) Two input streams as in Figure 2. (B) Their firing rates r ₁(t),r ₂(t). (C) Resulting firing activity of 100 neurons in the circuit. (D) Performance of a neural integrator, generated by feedback from a linear readout that was trained to output at any time t an approximation CA(t) of the integral over the difference of both input rates. Feedback values were injected as input currents into a randomly chosen subset of neurons in the circuit. Scale in nA shows average strength of feedback currents, also in (H). (E) Performance of linear readout that was trained to output 0 as long as CA(t) stayed below 0.83 nA, and to output r ₂(t) once CA(t) had crossed this threshold, as long as CA(t) stayed above 0.66 nA (i.e., in this test run during the shaded time periods). (F) Performance of linear readout trained to output r ₁(t) − CA(t), i.e., a combination of external and internal variables, at any time t (both r ₁ and CA normalized into the range [0,1]). (G) Response of a randomly chosen neuron in the circuit for ten repetitions of the same experiment (with input spike trains generated by Poisson processes with the same time course of firing rates), showing biologically realistic trial-to-trial variability. (H) Activity traces of a continuous attractor as in (D), but in eight different trials for eight different fixed values of r ₁ and r ₂ (shown on the right).

Figure 5. Organization of Input and Output Streams for the Three Computational Tasks Considered in the Computer Simulations

Each input stream consisted of multiple spike trains that provided synaptic inputs to individually chosen subsets of neurons in the recurrent circuit (which is indicated by a gray rectangle). (A,C) Input streams consisted of multiple Poisson spike trains with a time-varying firing rate r _i(t). (B) Input consisted of a burst (“cue”) in one spike train (which marks the beginning of a time interval) and independent Poisson spike train (“noise”) in the other input channels. (A–C) The actual outputs of the readouts (that were trained individually for each computational task) is shown in Figures 2–4.

Figure 6. Emulation of an FSM by a Noisy Fading-Memory System with Feedback According to Theorem 5

(A) Underlying open-loop system with noisy pattern detectors ₁, …, _k and suitable fading-memory readouts Ĥ ₁, …, Ĥ_l (which may also be subject to noise). (B) Resulting noise-robust emulation of an arbitrary given FSM by adding feedback to the system in (A). The same readouts as in (A) (denoted CL − Ĥ_j(t) in the closed loop) now encode the current state of the simulated FSM.

Figure 7. Evaluation of the Dependence of the Performance of the Circuit in Figure 4 on the Feedback Strength (i.e., on the Mean Amplitude of Current Injection from the Readout Back into Neurons in the Circuit)

For each feedback strength that was evaluated, the readouts were trained and tested for this feedback strength as for the preceding experiments. Error bars in (B–D) denote standard error. These control experiments show that the feedback is essential for the performance of the circuit.