Reinforcement Learning Using a Continuous Time Actor-Critic Framework with Spiking Neurons

Animals repeat rewarded behaviors, but the physiological basis of reward-based learning has only been partially elucidated. On one hand, experimental evidence shows that the neuromodulator dopamine carries information about rewards and affects synaptic plasticity. On the other hand, the theory of reinforcement learning provides a framework for reward-based learning. Recent models of reward-modulated spike-timing-dependent plasticity have made first steps towards bridging the gap between the two approaches, but faced two problems. First, reinforcement learning is typically formulated in a discrete framework, ill-adapted to the description of natural situations. Second, biologically plausible models of reward-modulated spike-timing-dependent plasticity require precise calculation of the reward prediction error, yet it remains to be shown how this can be computed by neurons. Here we propose a solution to these problems by extending the continuous temporal difference (TD) learning of Doya (2000) to the case of spiking neurons in an actor-critic network operating in continuous time, and with continuous state and action representations. In our model, the critic learns to predict expected future rewards in real time. Its activity, together with actual rewards, conditions the delivery of a neuromodulatory TD signal to itself and to the actor, which is responsible for action choice. In simulations, we show that such an architecture can solve a Morris water-maze-like navigation task, in a number of trials consistent with reported animal performance. We also use our model to solve the acrobot and the cartpole problems, two complex motor control tasks. Our model provides a plausible way of computing reward prediction error in the brain. Moreover, the analytically derived learning rule is consistent with experimental evidence for dopamine-modulated spike-timing-dependent plasticity.

Models of Reward-Modulated Spike-Timing-Dependent Plasticity

Nicolas Frémaux

How do animals learn to repeat behaviors that lead to the obtention of food or other “rewarding” objects? As a biologically plausible paradigm for learning in spiking neural networks, spike-timing dependent plasticity (STDP) has been shown to perform well in unsupervised learning tasks such as receptive field development. However, STDP fails to take behavioral relevance into account, and as such is inadequate to explain a vast range of learning tasks in which the final outcome, conditioned on the prior execution of a series of actions, is signaled to an animal through sparse rewards. In this thesis, I show that the addition of a third, global, reward-based factor to the pre- and postsynaptic factors of STDP is a promising solution to this problem, consistent with experimental findings. One one hand, dopamine is a neuromodulator which has been shown to encode reward signals in the brain. On the other hand, STDP has been shown to be affected by dopamine, even though the precise nature of the interaction is unclear. Moreover, the theoretical framework of reinforcement learning provides strong foundation for the analysis of these learning rules. After studying existing examples of such rules in a navigation task, I derive simple functional requirements for reward-modulated learning rules, and illustrate these in a motor learning task. One of those functional requirements is the existence a “critic” structure, constantly evaluating the potential for rewarding events. The implication of the existence of such a critic on the interpretation of psychophysical experiments are also discussed. Finally, I propose a biologically plausible implementation of such a structure, that performs motor or navigational tasks. This is based on a generalization of temporal difference learning, a well-known reinforcement learning framework, to continuous time, well suited to an implementation with spiking neurons. These result provide a unified picture of reward-modulated learning rules: even though different rules have been proposed, these can be reduced to a single model at the synaptic level, with variations in the computation of the neuromodulatory signal enabling switching between different learning rules.

EPFL2013

Spike-Based Reinforcement Learning in Continuous State and Action Space: When Policy Gradient Methods Fail

Nicolas Frémaux, Wulfram Gerstner, Walter Senn, Eleni Vasilaki

Changes of synaptic connections between neurons are thought to be the physiological basis of learning. These changes can be gated by neuromodulators that encode the presence of reward. We study a family of reward-modulated synaptic learning rules for spiking neurons on a learning task in continuous space inspired by the Morris Water maze. The synaptic update rule modifies the release probability of synaptic transmission and depends on the timing of presynaptic spike arrival, postsynaptic action potentials, as well as the membrane potential of the postsynaptic neuron. The family of learning rules includes an optimal rule derived from policy gradient methods as well as reward modulated Hebbian learning. The synaptic update rule is implemented in a population of spiking neurons using a network architecture that combines feedforward input with lateral connections. Actions are represented by a population of hypothetical action cells with strong mexican-hat connectivity and are read out at theta frequency. We show that in this architecture, a standard policy gradient rule fails to solve the Morris watermaze task, whereas a variant with a Hebbian bias can learn the task within 20 trials, consistent with experiments. This result does not depend on implementation details such as the size of the neuronal populations. Our theoretical approach shows how learning new behaviors can be linked to reward-modulated plasticity at the level of single synapses and makes predictions about the voltage and spike-timing dependence of synaptic plasticity and the influence of neuromodulators such as dopamine. It is an important step towards connecting formal theories of reinforcement learning with neuronal and synaptic properties

Public Library of Science2009

Prospective Coding by Spiking Neurons

Johanni Michael Brea, Walter Senn

Animals learn to make predictions, such as associating the sound of a bell with upcoming feeding or predicting a movement that a motor command is eliciting. How predictions are realized on the neuronal level and what plasticity rule underlies their learning is not well understood. Here we propose a biologically plausible synaptic plasticity rule to learn predictions on a single neuron level on a timescale of seconds. The learning rule allows a spiking two-compartment neuron to match its current firing rate to its own expected future discounted firing rate. For instance, if an originally neutral event is repeatedly followed by an event that elevates the firing rate of a neuron, the originally neutral event will eventually also elevate the neuron's firing rate. The plasticity rule is a form of spike timing dependent plasticity in which a presynaptic spike followed by a postsynaptic spike leads to potentiation. Even if the plasticity window has a width of 20 milliseconds, associations on the time scale of seconds can be learned. We illustrate prospective coding with three examples: learning to predict a time varying input, learning to predict the next stimulus in a delayed paired-associate task and learning with a recurrent network to reproduce a temporally compressed version of a sequence. We discuss the potential role of the learning mechanism in classical trace conditioning. In the special case that the signal to be predicted encodes reward, the neuron learns to predict the discounted future reward and learning is closely related to the temporal difference learning algorithm TD(lambda).

Public Library of Science2016

Models of Reward-Modulated Spike-Timing-Dependent Plasticity

Nicolas Frémaux

EPFL2013