A fundamental function of cortical circuits is the integration of information from different sources to form a reliable basis for behavior. While animals behave as if they optimally integrate information according to Bayesian probability theory, the implementation of the required computations in the biological substrate remains unclear. We propose a novel, Bayesian view on the dynamics of conductance-based neurons and synapses which suggests that they are naturally equipped to optimally perform information integration. In our approach apical dendrites represent prior expectations over somatic potentials, while basal dendrites represent likelihoods of somatic potentials. These are parametrized by local quantities, the effective reversal potentials and membrane conductances. We formally demonstrate that under these assumptions the somatic compartment naturally computes the corresponding posterior. We derive a gradient-based plasticity rule, allowing neurons to learn desired target distributions and weight synaptic inputs by their relative reliabilities. Our theory explains various experimental findings on the system and single-cell level related to multi-sensory integration, which we illustrate with simulations. Furthermore, we make experimentally testable predictions on Bayesian dendritic integration and synaptic plasticity.

Being permanently confronted with an uncertain world, brains have faced evolutionary pressure to represent this uncertainty in order to respond appropriately. Often, this requires visiting multiple interpretations of the available information or multiple solutions to an encountered problem. This gives rise to the so-called mixing problem: since all of these "valid" states represent powerful attractors, but between themselves can be very dissimilar, switching between such states can be difficult. We propose that cortical oscillations can be effectively used to overcome this challenge. By acting as an effective temperature, background spiking activity modulates exploration. Rhythmic changes induced by cortical oscillations can then be interpreted as a form of simulated tempering. We provide a rigorous mathematical discussion of this link and study some of its phenomenological implications in computer simulations. This identifies a new computational role of cortical oscillations and connects them to various phenomena in the brain, such as sampling-based probabilistic inference, memory replay, multisensory cue combination, and place cell flickering.

We present first experimental results on the novel BrainScaleS-2 neuromorphic architecture based on an analog neuro-synaptic core and augmented by embedded microprocessors for complex plasticity and experiment control. The high acceleration factor of 1000 compared to biological dynamics enables the execution of computationally expensive tasks, by allowing the fast emulation of long-duration experiments or rapid iteration over many consecutive trials. The flexibility of our architecture is demonstrated in a suite of five distinct experiments, which emphasize different aspects of the BrainScaleS-2 system.

We demonstrate five experiments that are performed on the novel BrainScaleS-2 neuromorphic architecture based on an analog neuro-synaptic core and augmented by embedded microprocessors for complex plasticity and experiment control. The high acceleration factor of 1000 compared to biological dynamics enables the execution of computationally expensive tasks, by allowing the fast emulation of long-duration experiments or rapid iteration over many consecutive trials. The experiments were chosen to emphasize different aspects of the BrainScaleS-2 system, especially involving on-chip plasticity and closed-loop scenarios.

An increasing body of evidence suggests that the trial-to-trial variability of spiking activity in the brain is not mere noise, but rather the reflection of a sampling-based encoding scheme for probabilistic computing. Since the precise statistical properties of neural activity are important in this context, many models assume an ad-hoc source of well-behaved, explicit noise, either on the input or on the output side of single neuron dynamics, most often assuming an independent Poisson process in either case. However, these assumptions are somewhat problematic: neighboring neurons tend to share receptive fields, rendering both their input and their output correlated; at the same time, neurons are known to behave largely deterministically, as a function of their membrane potential and conductance. We suggest that spiking neural networks may have no need for noise to perform sampling-based Bayesian inference. We study analytically the effect of auto- and cross-correlations in functional Bayesian spiking networks and demonstrate how their effect translates to synaptic interaction strengths, rendering them controllable through synaptic plasticity. This allows even small ensembles of interconnected deterministic spiking networks to simultaneously and co-dependently shape their output activity through learning, enabling them to perform complex Bayesian computation without any need for noise, which we demonstrate in silico, both in classical simulation and in neuromorphic emulation. These results close a gap between the abstract models and the biology of functionally Bayesian spiking networks, effectively reducing the architectural constraints imposed on physical neural substrates required to perform probabilistic computing, be they biological or artificial.

Deep learning has seen remarkable developments over the last years, many of them inspired by neuroscience. However, the main learning mechanism behind these advances – error backpropagation – appears to be at odds with neurobiology. Here, we introduce a multilayer neuronal network model with simplified dendritic compartments in which error-driven synaptic plasticity adapts the network towards a global desired output. In contrast to previous work our model does not require separate phases and synaptic learning is driven by local dendritic prediction errors continuously in time. Such errors originate at apical dendrites and occur due to a mismatch between predictive input from lateral interneurons and activity from actual top-down feedback. Through the use of simple dendritic compartments and different cell-types our model can represent both error and normal activity within a pyramidal neuron. We demonstrate the learning capabilities of the model in regression and classification tasks, and show analytically that it approximates the error backpropagation algorithm. Moreover, our framework is consistent with recent observations of learning between brain areas and the architecture of cortical microcircuits. Overall, we introduce a novel view of learning on dendritic cortical circuits and on how the brain may solve the long-standing synaptic credit assignment problem.

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.

Humans and animals face decision tasks in an uncertain multi-agent environment where an agent's strategy may change in time due to the co-adaptation of others strategies. The neuronal substrate and the computational algorithms underlying such adaptive decision making, however, is largely unknown. We propose a population coding model of spiking neurons with a policy gradient procedure that successfully acquires optimal strategies for classical game-theoretical tasks. The suggested population reinforcement learning reproduces data from human behavioral experiments for the blackjack and the inspector game. It performs optimally according to a pure (deterministic) and mixed (stochastic) Nash equilibrium, respectively. In contrast, temporal-difference(TD)-learning, covariance-learning, and basic reinforcement learning fail to perform optimally for the stochastic strategy. Spike-based population reinforcement learning, shown to follow the stochastic reward gradient, is therefore a viable candidate to explain automated decision learning of a Nash equilibrium in two-player games.

We consider a statistical framework in which recurrent networks of spiking neurons learn to generate spatio-temporal spike patterns. Given biologically realistic stochastic neuronal dynamics we derive a tractable learning rule for the synaptic weights towards hidden and visible neurons that leads to optimal recall of the training sequences. We show that learning synaptic weights towards hidden neurons significantly improves the storing capacity of the network. Furthermore, we derive an approximate online learning rule and show that our learning rule is consistent with Spike-Timing Dependent Plasticity in that if a presynaptic spike shortly precedes a postynaptic spike, potentiation is induced and otherwise depression is elicited.

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.

The neuronal processing of visual stimuli in primary visual cortex (V1) can be modified by perceptual training. Training in bisection discrimination, for instance, changes the contextual interactions in V1 elicited by parallel lines. Before training, two parallel lines inhibit their individual V1-responses. After bisection training, inhibition turns into non-symmetric excitation while performing the bisection task. Yet, the receptive field of the V1 neurons evaluated by a single line does not change during task performance. We present a model of recurrent processing in V1 where the neuronal gain can be modulated by a global attentional signal. Perceptual learning mainly consists in strengthening this attentional signal, leading to a more effective gain modulation. The model reproduces both the psychophysical results on bisection learning and the modified contextual interactions observed in V1 during task performance. It makes several predictions, for instance that imagery training should improve the performance, or that a slight stimulus wiggling can strongly affect the representation in V1 while performing the task. We conclude that strengthening a top-down induced gain increase can explain perceptual learning, and that this top-down signal can modify lateral interactions within V1, without significantly changing the classical receptive field of V1 neurons.

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.

We present a model of spike-driven synaptic plasticity inspired by experimental observations and motivated by the desire to build an electronic hardware device that can learn to classify complex stimuli in a semisupervised fashion. During training, patterns of activity are sequentially imposed on the input neurons, and an additional instructor signal drives the output neurons toward the desired activity. The network is made of integrate-and-fire neurons with constant leak and a floor. The synapses are bistable, and they are modified by the arrival of presynaptic spikes. The sign of the change is determined by both the depolarization and the state of a variable that integrates the postsynaptic action potentials. Following the training phase, the instructor signal is removed, and the output neurons are driven purely by the activity of the input neurons weighted by the plastic synapses. In the absence of stimulation, the synapses preserve their internal state indefinitely. Memories are also very robust to the disruptive action of spontaneous activity. A network of 2000 input neurons is shown to be able to classify correctly a large number (thousands) of highly overlapping patterns (300 classes of preprocessed Latex characters, 30 patterns per class, and a subset of the NIST characters data set) and to generalize with performances that are better than or comparable to those of artificial neural networks. Finally we show that the synaptic dynamics is compatible with many of the experimental observations on the induction of long-term modifications (spike-timing-dependent plasticity and its dependence on both the postsynaptic depolarization and the frequency of pre- and postsynaptic neurons).

The primary visual cortex (V1) is pre-wired to facilitate the extraction of behaviorally important visual features. Collinear edge detectors in V1, for instance, mutually enhance each other to improve the perception of lines against a noisy background. The same pre-wiring that facilitates line extraction, however, is detrimental when subjects have to discriminate the brightness of different line segments. How is it possible to improve in one task by unsupervised practicing, without getting worse in the other task? The classical view of perceptual learning is that practicing modulates the feedforward input stream through synaptic modifications onto or within V1. However, any rewiring of V1 would deteriorate other perceptual abilities different from the trained one. We propose a general neuronal model showing that perceptual learning can modulate top-down input to V1 in a task-specific way while feedforward and lateral pathways remain intact. Consistent with biological data, the model explains how context-dependent brightness discrimination is improved by a top-down recruitment of recurrent inhibition and a top-down induced increase of the neuronal gain within V1. Both the top-down modulation of inhibition and of neuronal gain are suggested to be universal features of cortical microcircuits which enable perceptual learning.

Learning in a neuronal network is often thought of as a linear superposition of synaptic modifications induced by individual stimuli. However, since biological synapses are naturally bounded, a linear superposition would cause fast forgetting of previously acquired memories. Here we show that this forgetting can be avoided by introducing additional constraints on the synaptic and neural dynamics. We consider Hebbian plasticity of excitatory synapses. A synapse is modified only if the postsynaptic response does not match the desired output. With this learning rule, the original memory performances with unbounded weights are regained, provided that (1) there is some global inhibition, (2) the learning rate is small, and (3) the neurons can discriminate small differences in the total synaptic input (e.g., by making the neuronal threshold small compared to the total postsynaptic input). We prove in the form of a generalized perceptron convergence theorem that under these constraints, a neuron learns to classify any linearly separable set of patterns, including a wide class of highly correlated random patterns. During the learning process, excitation becomes roughly balanced by inhibition, and the neuron classifies the patterns on the basis of small differences around this balance. The fact that synapses saturate has the additional benefit that nonlinearly separable patterns, such as similar patterns with contradicting outputs, eventually generate a subthreshold response, and therefore silence neurons that cannot provide any information.

Rate models are often used to study the behavior of large networks of spiking neurons. Here we propose a procedure to derive rate models that take into account the fluctuations of the input current and firing-rate adaptation, two ubiquitous features in the central nervous system that have been previously overlooked in constructing rate models. The procedure is general and applies to any model of firing unit. As examples, we apply it to the leaky integrate-and-fire (IF) neuron, the leaky IF neuron with reversal potentials, and to the quadratic IF neuron. Two mechanisms of adaptation are considered, one due to an afterhyperpolarization current and the other to an adapting threshold for spike emission. The parameters of these simple models can be tuned to match experimental data obtained from neocortical pyramidal neurons. Finally, we show how the stationary model can be used to predict the time-varying activity of a large population of adapting neurons.

Systematic temporal relations between single neuronal activities or population activities are ubiquitous in the brain. No experimental evidence, however, exists for a direct modification of neuronal delays during Hebbian-type stimulation protocols. We show that in fact an explicit delay adaptation is not needed if one assumes that the synaptic strengths are modified according to the recently observed temporally asymmetric learning rule with the downregulating branch dominating the upregulating branch. During development, slow, unbiased fluctuations in the transmission time, together with temporally correlated network activity, may control neural growth and implicitly induce drifts in the axonal delays and dendritic latencies. These delays and latencies become optimally tuned in the sense that the synaptic response tends to peak in the soma of the postsynaptic cell if this is most likely to fire. The nature of the selection process requires unreliable synapses in order to give successful synapses an evolutionary advantage over the others. The width of the learning function also determines the preferred dendritic delay and the preferred width of the postsynaptic response. Hence, it may implicitly determine whether a synaptic connection provides a precisely timed or a broadly tuned "contextual" signal.

The precise times of occurrence of individual pre- and postsynaptic action potentials are known to play a key role in the modification of synaptic efficacy. Based on stimulation protocols of two synaptically connected neurons, we infer an algorithm that reproduces the experimental data by modifying the probability of vesicle discharge as a function of the relative timing of spikes in the pre- and postsynaptic neurons. The primary feature of this algorithm is an asymmetry with respect to the direction of synaptic modification depending on whether the presynaptic spikes precede or follow the postsynaptic spike. Specifically, if the presynaptic spike occurs up to 50 ms before the postsynaptic spike, the probability of vesicle discharge is upregulated, while the probability of vesicle discharge is downregulated if the presynaptic spike occurs up to 50 ms after the postsynaptic spike. When neurons fire irregularly with Poisson spike trains at constant mean firing rates, the probability of vesicle discharge converges toward a characteristic value determined by the pre- and postsynaptic firing rates. On the other hand, if the mean rates of the Poisson spike trains slowly change with time, our algorithm predicts modifications in the probability of release that generalize Hebbian and Bienenstock-Cooper-Munro rules. We conclude that the proposed spike-based synaptic learning algorithm provides a general framework for regulating neurotransmitter release probability.

A recent experiment showed that neurons in the primary auditory cortex of the monkey do not change their mean firing rate during an ongoing tone stimulus. The only change was an enhanced correlation among the individual spike trains during the tone. We show that there is an easy way to extract this coherence information in the cortical cell population by projecting the spike trains through depressing synapses onto a postsynaptic neuron.

Numerous animal behaviors, such as locomotion in vertebrates, are produced by rhythmic contractions that alternate between two muscle groups. The neuronal networks generating such alternate rhythmic activity are generally thought to rely on pacemaker cells or well-designed circuits consisting of inhibitory and excitatory neurons. However, experiments in organotypic cultures of embryonic rat spinal cord have shown that neuronal networks with purely excitatory and random connections may oscillate due to their synaptic depression, even without pacemaker cells. In this theoretical study, we investigate what happens if two such networks are symmetrically coupled by a small number of excitatory connections. We discuss a time-discrete mean-field model describing the average activity and the average synaptic depression of the two networks. Depending on the parameter values of the depression, the oscillations will be in phase, antiphase, quasiperiodic, or phase trapped. We put forward the hypothesis that pattern generators may rely on activity-dependent tuning of synaptic depression.