Statistical physics and practical training of soft-committee machines

Equilibrium states of large layered neural networks with differentiable activation function and a single, linear output unit are investigated using the replica formalism. The quenched free energy of a student network with a very large number of hidden units learning a rule of perfectly matching complexity is calculated analytically. The system undergoes a first order phase transition from unspecialized to specialized student configurations at a critical size of the training set. Computer simulations of learning by stochastic gradient descent from a fixed training set demonstrate that the equilibrium results describe quantitatively the plateau states which occur in practical training procedures at sufficiently small but finite learning rates.Comment: 11 pages, 4 figure

SNA – a toolbox for the stoichiometric analysis of metabolic networks

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: . 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

Stimulus sampling as an exploration mechanism for fast reinforcement learning

Reinforcement learning in neural networks requires a mechanism for exploring new network states in response to a single, nonspecific reward signal. Existing models have introduced synaptic or neuronal noise to drive this exploration. However, those types of noise tend to almost average out—precluding or significantly hindering learning —when coding in neuronal populations or by mean firing rates is considered. Furthermore, careful tuning is required to find the elusive balance between the often conflicting demands of speed and reliability of learning. Here we show that there is in fact no need to rely on intrinsic noise. Instead, ongoing synaptic plasticity triggered by the naturally occurring online sampling of a stimulus out of an entire stimulus set produces enough fluctuations in the synaptic efficacies for successful learning. By combining stimulus sampling with reward attenuation, we demonstrate that a simple Hebbian-like learning rule yields the performance that is very close to that of primates on visuomotor association tasks. In contrast, learning rules based on intrinsic noise (node and weight perturbation) are markedly slower. Furthermore, the performance advantage of our approach persists for more complex tasks and network architectures. We suggest that stimulus sampling and reward attenuation are two key components of a framework by which any single-cell supervised learning rule can be converted into a reinforcement learning rule for networks without requiring any intrinsic noise sourc

Reinforcement learning in dendritic structures

The discovery of binary dendritic events such as local NMDA spikes in dendritic subbranches led to the suggestion that dendritic trees could be computationally equivalent to a 2-layer network of point neurons, with a single output unit represented by the soma, and input units represented by the dendritic branches. Although this interpretation endows a neuron with a high computational power, it is functionally not clear why nature would have preferred the dendritic solution with a single but complex neuron, as opposed to the network solution with many but simple units. We show that the dendritic solution has a distinguished advantage over the network solution when considering different learning tasks. Its key property is that the dendritic branches receive an immediate feedback from the somatic output spike, while in the corresponding network architecture the feedback would require additional backpropagating connections to the input units. Assuming a reinforcement learning scenario we formally derive a learning rule for the synaptic contacts on the individual dendritic trees which depends on the presynaptic activity, the local NMDA spikes, the somatic action potential, and a delayed reinforcement signal. We test the model for two scenarios: the learning of binary classifications and of precise spike timings. We show that the immediate feedback represented by the backpropagating action potential supplies the individual dendritic branches with enough information to efficiently adapt their synapses and to speed up the learning process

Reinforcement learning in populations of spiking neurons

Population coding is widely regarded as a key mechanism for achieving reliable behavioral responses in the face of neuronal variability. But in standard reinforcement learning a flip-side becomes apparent. Learning slows down with increasing population size since the global reinforcement becomes less and less related to the performance of any single neuron. We show that, in contrast, learning speeds up with increasing population size if feedback about the populationresponse modulates synaptic plasticity in addition to global reinforcement. The two feedback signals (reinforcement and population-response signal) can be encoded by ambient neurotransmitter concentrations which vary slowly, yielding a fully online plasticity rule where the learning of a stimulus is interleaved with the processing of the subsequent one. The assumption of a single additional feedback mechanism therefore reconciles biological plausibility with efficient learning

Stimulus sampling as an exploration mechanism for fast reinforcement learning

Reinforcement learning in neural networks requires a mechanism for exploring new network states in response to a single, nonspecific reward signal. Existing models have introduced synaptic or neuronal noise to drive this exploration. However, those types of noise tend to almost average out—precluding or significantly hindering learning —when coding in neuronal populations or by mean firing rates is considered. Furthermore, careful tuning is required to find the elusive balance between the often conflicting demands of speed and reliability of learning. Here we show that there is in fact no need to rely on intrinsic noise. Instead, ongoing synaptic plasticity triggered by the naturally occurring online sampling of a stimulus out of an entire stimulus set produces enough fluctuations in the synaptic efficacies for successful learning. By combining stimulus sampling with reward attenuation, we demonstrate that a simple Hebbian-like learning rule yields the performance that is very close to that of primates on visuomotor association tasks. In contrast, learning rules based on intrinsic noise (node and weight perturbation) are markedly slower. Furthermore, the performance advantage of our approach persists for more complex tasks and network architectures. We suggest that stimulus sampling and reward attenuation are two key components of a framework by which any single-cell supervised learning rule can be converted into a reinforcement learning rule for networks without requiring any intrinsic noise source

Scale-Free Navigational Planning by Neuronal Traveling Waves

Spatial navigation and planning is assumed to involve a cognitive map for evaluating trajectories towards a goal. How such a map is realized in neuronal terms, however, remains elusive. Here we describe a simple and noise-robust neuronal implementation of a path finding algorithm in complex environments. We consider a neuronal map of the environment that supports a traveling wave spreading out from the goal location opposite to direction of the physical movement. At each position of the map, the smallest firing phase between adjacent neurons indicate the shortest direction towards the goal. In contrast to diffusion or single-wave-fronts, local phase differences build up in time at arbitrary distances from the goal, providing a minimal and robust directional information throughout the map. The time needed to reach the steady state represents an estimate of an agent's waiting time before it heads off to the goal. Given typical waiting times we estimate the minimal number of neurons involved in the cognitive map. In the context of the planning model, forward and backward spread of neuronal activity, oscillatory waves, and phase precession get a functional interpretation, allowing for speculations about the biological counterpart

Prospective Coding by Spiking Neurons

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)