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(λ).

In the last decade dendrites of cortical neurons have been shown to nonlinearly combine synaptic inputs by evoking local dendritic spikes. It has been suggested that these nonlinearities raise the computational power of a single neuron, making it comparable to a 2-layer network of point neurons. But how these nonlinearities can be incorporated into the synaptic plasticity to optimally support learning remains unclear. We present a theoretically derived synaptic plasticity rule for supervised and reinforcement learning that depends on the timing of the presynaptic, the dendritic and the postsynaptic spikes. For supervised learning, the rule can be seen as a biological version of the classical error-backpropagation algorithm applied to the dendritic case. When modulated by a delayed reward signal, the same plasticity is shown to maximize the expected reward in reinforcement learning for various coding scenarios. Our framework makes specific experimental predictions and highlights the unique advantage of active dendrites for implementing powerful synaptic plasticity rules that have access to downstream information via backpropagation of action potentials.

Population coding is widely regarded as a key mechanism for achieving reliable behavioral decisions. We previously introduced reinforcement learning for population-based decision making by spiking neurons. Here we generalize population reinforcement learning to spike-based plasticity rules that take account of the postsynaptic neural code. We consider spike/no-spike, spike count and spike latency codes. The multi-valued and continuous-valued features in the postsynaptic code allow for a generalization of binary decision making to multi-valued decision making and continuous-valued action selection. We show that code-specific learning rules speed up learning both for the discrete classification and the continuous regression tasks. The suggested learning rules also speed up with increasing population size as opposed to standard reinforcement learning rules. Continuous action selection is further shown to explain realistic learning speeds in the Morris water maze. Finally, we introduce the concept of action perturbation as opposed to the classical weight- or node-perturbation as an exploration mechanism underlying reinforcement learning. Exploration in the action space greatly increases the speed of learning as compared to exploration in the neuron or weight space.

Recent experiments revealed that the fruit fly Drosophila melanogaster has a dedicated mechanism for forgetting: blocking the G-protein Rac leads to slower and activating Rac to faster forgetting. This active form of forgetting lacks a satisfactory functional explanation. We investigated optimal decision making for an agent adapting to a stochastic environment where a stimulus may switch between being indicative of reward or punishment. Like Drosophila, an optimal agent shows forgetting with a rate that is linked to the time scale of changes in the environment. Moreover, to reduce the odds of missing future reward, an optimal agent may trade the risk of immediate pain for information gain and thus forget faster after aversive conditioning. A simple neuronal network reproduces these features. Our theory shows that forgetting in Drosophila appears as an optimal adaptive behavior in a changing environment. This is in line with the view that forgetting is adaptive rather than a consequence of limitations of the memory system.

In learning from trial and error, animals need to relate behavioral decisions to environmental reinforcement even though it may be difficult to assign credit to a particular decision when outcomes are uncertain or subject to delays. When considering the biophysical basis of learning, the credit-assignment problem is compounded because the behavioral decisions themselves result from the spatio-temporal aggregation of many synaptic releases. We present a model of plasticity induction for reinforcement learning in a population of leaky integrate and fire neurons which is based on a cascade of synaptic memory traces. Each synaptic cascade correlates presynaptic input first with postsynaptic events, next with the behavioral decisions and finally with external reinforcement. For operant conditioning, learning succeeds even when reinforcement is delivered with a delay so large that temporal contiguity between decision and pertinent reward is lost due to intervening decisions which are themselves subject to delayed reinforcement. This shows that the model provides a viable mechanism for temporal credit assignment. Further, learning speeds up with increasing population size, so the plasticity cascade simultaneously addresses the spatial problem of assigning credit to synapses in different population neurons. Simulations on other tasks, such as sequential decision making, serve to contrast the performance of the proposed scheme to that of temporal difference-based learning. We argue that, due to their comparative robustness, synaptic plasticity cascades are attractive basic models of reinforcement learning in the brain.

We investigate a recently proposed model for decision learning in a population of spiking neurons where synaptic plasticity is modulated by a population signal in addition to reward feedback. For the basic model, binary population decision making based on spike/no-spike coding, a detailed computational analysis is given about how learning performance depends on population size and task complexity. Next, we extend the basic model to n-ary decision making and show that it can also be used in conjunction with other population codes such as rate or even latency coding.

We introduce a new supervised learning rule for the tempotron task: the binary classification of input spike trains by an integrate-and-fire neuron that encodes its decision by firing or not firing. The rule is based on the gradient of a cost function, is found to have enhanced performance, and does not rely on a specific reset mechanism in the integrate-and-fire neuron.

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.

BACKGROUND: Despite recent algorithmic and conceptual progress, the stoichiometric network analysis of large metabolic models remains a computationally challenging problem. RESULTS: SNA is a interactive, high performance toolbox for analysing the possible steady state behaviour of metabolic networks by computing the generating and elementary vectors of their flux and conversions cones. It also supports analysing the steady states by linear programming. The toolbox is implemented mainly in Mathematica and returns numerically exact results. It is available under an open source license from: http://bioinformatics.org/project/?group_id=546. CONCLUSION: Thanks to its performance and modular design, SNA is demonstrably useful in analysing genome scale metabolic networks. Further, the integration into Mathematica provides a very flexible environment for the subsequent analysis and interpretation of the results.

MOTIVATION: Genome scale analysis of the metabolic network of a microorganism is a major challenge in bioinformatics. The combinatorial explosion, which occurs during the construction of elementary fluxes (non-redundant pathways) requires sophisticated and efficient algorithms to tackle the problem. RESULTS: Mathematically, the calculation of elementary fluxes amounts to characterizing the space of solutions to a mixed system of linear equalities, given by the stoichiometry matrix, and linear inequalities, arising from the irreversibility of some or all of the reactions in the network. Previous approaches to this problem have iteratively solved for the equalities while satisfying the inequalities throughout the process. In an extension of previous work, here we consider the complementary approach and derive an algorithm which satisfies the inequalities one by one while staying in the space of solution of the equality constraints. Benchmarks on different subnetworks of the central carbon metabolism of Escherichia coli show that this new approach yields a significant reduction in the execution time of the calculation. This reduction arises since the odds that an intermediate elementary flux already fulfills an additional inequality are larger than when having to satisfy an additional equality constraint.

MOTIVATION: An important tool in Systems Biology is the stoichiometric modeling of metabolic networks, where the stationary states of the network are described by a high-dimensional polyhedral cone, the so-called flux cone. Exhaustive descriptions of the metabolism can be obtained by computing the elementary vectors of this cone but, owing to a combinatorial explosion of the number of elementary vectors, this approach becomes computationally intractable for genome scale networks. RESULT: Hence, we propose to instead focus on the conversion cone, a projection of the flux cone, which describes the interaction of the metabolism with its external chemical environment. We present a direct method for calculating the elementary vectors of this cone and, by studying the metabolism of Saccharomyces cerevisiae, we demonstrate that such an analysis is computationally feasible even for genome scale networks.

Using methods of Statistical Physics, we investigate the rOle of model complexity in learning with support vector machines (SVMs). We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal general(cid:173) ization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which only depend on the target rule but not on the SVM kernel.

Statistical mechanics is used to study generalization in a tree committee machine with K hidden units and continuous weights trained on examples generated by a teacher of the same structure but corrupted by noise. The corruption is due to additive gaussian noise applied in the input layer or the hidden layer of the teacher. In the large K limit the generalization error εg as function of α, the number of patterns per adjustable parameter, shows a qualitatively similar behavior for the two cases: It does not approach its optimal value and is nonmonotonic if training is done at zero temperature. This remains true even when replica symmetry breaking is taken into account. Training at a fixed positive temperature leads, within the replica symmetric theory, to an α-k decay of εg toward its optimal value. The value of k is calculated and found to depend on the model of noise. By scaling the temperature with α, the value of k can be increased to an optimal value kopt. However, at one step of replica symmetry breaking at a fixed positive temperature εg decays as α−kopt. So, although εg will approach its optimal value with increasing sample size for any fixed K, the convergence is only uniform in K when training at a positive temperature.