Theoretical neuroscientists and machine learning researchers have proposed a variety of learning rules to enable artificial neural networks to effectively perform both supervised and unsupervised learning tasks. It is not always clear, however, how these theoretically-derived rules relate to biological mechanisms of plasticity in the brain, or how these different rules might be mechanistically implemented in different contexts and brain regions. This study shows that the calcium control hypothesis, which relates synaptic plasticity in the brain to the calcium concentration ([Ca2+]) in dendritic spines, can produce a diverse array of learning rules. We propose a simple, perceptron-like neuron model, the calcitron, that has four sources of [Ca2+]: local (following the activation of an excitatory synapse and confined to that synapse), heterosynaptic (resulting from the activity of other synapses), postsynaptic spike-dependent, and supervisor-dependent. We demonstrate that by modulating the plasticity thresholds and calcium influx from each calcium source, we can reproduce a wide range of learning and plasticity protocols, such as Hebbian and anti-Hebbian learning, frequency-dependent plasticity, and unsupervised recognition of frequently repeating input patterns. Moreover, by devising simple neural circuits to provide supervisory signals, we show how the calcitron can implement homeostatic plasticity, perceptron learning, and BTSP-inspired one-shot learning. Our study bridges the gap between theoretical learning algorithms and their biological counterparts, not only replicating established learning paradigms but also introducing novel rules.

The output of neocortical layer 5 pyramidal cells (L5PCs) is expressed by a train of single spikes with intermittent bursts of multiple spikes at high frequencies. The bursts are the result of nonlinear dendritic properties, including Na+, Ca2+, and NMDA spikes, that interact with the ~10,000 synapses impinging on the neuron's dendrites. Output spike bursts are thought to implement key dendritic computations, such as coincidence detection of bottom-up inputs (arriving mostly at the basal tree) and top-down inputs (arriving mostly at the apical tree). In this study we used a detailed nonlinear model of L5PC receiving excitatory and inhibitory synaptic inputs to explore the conditions for generating bursts and for modulating their properties. We established the excitatory input conditions on the basal versus the apical tree that favor burst and show that there are two distinct types of bursts. Bursts consisting of 3 or more spikes firing at < 200 Hz, which are generated by stronger excitatory input to the basal versus the apical tree, and bursts of ~2-spikes at ~250 Hz, generated by prominent apical tuft excitation. Localized and well-timed dendritic inhibition on the apical tree differentially modulates Na+, Ca2+, and NMDA spikes and, consequently, finely controls the burst output. Finally, we explored the implications of different burst classes and respective dendritic inhibition for regulating synaptic plasticity.

Synaptic clustering on neuronal dendrites has been hypothesized to play an important role in implementing pattern recognition. Neighboring synapses on a dendritic branch can interact in a synergistic, cooperative manner via nonlinear voltage-dependent mechanisms, such as NMDA receptors. Inspired by the NMDA receptor, the single-branch clusteron learning algorithm takes advantage of location-dependent multiplicative nonlinearities to solve classification tasks by randomly shuffling the locations of "under-performing" synapses on a model dendrite during learning ("structural plasticity"), eventually resulting in synapses with correlated activity being placed next to each other on the dendrite. We propose an alternative model, the gradient clusteron, or G-clusteron, which uses an analytically-derived gradient descent rule where synapses are "attracted to" or "repelled from" each other in an input- and location-dependent manner. We demonstrate the classification ability of this algorithm by testing it on the MNIST handwritten digit dataset and show that, when using a softmax activation function, the accuracy of the G-clusteron on the all-versus-all MNIST task (~85%) approaches that of logistic regression (~93%). In addition to the location update rule, we also derive a learning rule for the synaptic weights of the G-clusteron ("functional plasticity") and show that a G-clusteron that utilizes the weight update rule can achieve ~89% accuracy on the MNIST task. We also show that a G-clusteron with both the weight and location update rules can learn to solve the XOR problem from arbitrary initial conditions.

Retinal direction-selectivity originates in starburst amacrine cells (SACs), which display a centrifugal preference, responding with greater depolarization to a stimulus expanding from soma to dendrites than to a collapsing stimulus. Various mechanisms were hypothesized to underlie SAC centrifugal preference, but dissociating them is experimentally challenging and the mechanisms remain debatable. To address this issue, we developed the Retinal Stimulation Modeling Environment (RSME), a multifaceted data-driven retinal model that encompasses detailed neuronal morphology and biophysical properties, retina-tailored connectivity scheme and visual input. Using a genetic algorithm, we demonstrated that spatiotemporally diverse excitatory inputs-sustained in the proximal and transient in the distal processes-are sufficient to generate experimentally validated centrifugal preference in a single SAC. Reversing these input kinetics did not produce any centrifugal-preferring SAC. We then explored the contribution of SAC-SAC inhibitory connections in establishing the centrifugal preference. SAC inhibitory network enhanced the centrifugal preference, but failed to generate it in its absence. Embedding a direction selective ganglion cell (DSGC) in a SAC network showed that the known SAC-DSGC asymmetric connectivity by itself produces direction selectivity. Still, this selectivity is sharpened in a centrifugal-preferring SAC network. Finally, we use RSME to demonstrate the contribution of SAC-SAC inhibitory connections in mediating direction selectivity and recapitulate recent experimental findings. Thus, using RSME, we obtained a mechanistic understanding of SACs' centrifugal preference and its contribution to direction selectivity.

The visual system must make predictions to compensate for inherent delays in its processing. Yet little is known, mechanistically, about how prediction aids natural behaviors. Here, we show that despite a 20-30ms intrinsic processing delay, the vertical motion sensitive (VS) network of the blowfly achieves maximally efficient prediction. This prediction enables the fly to fine-tune its complex, yet brief, evasive flight maneuvers according to its initial ego-rotation at the time of detection of the visual threat. Combining a rich database of behavioral recordings with detailed compartmental modeling of the VS network, we further show that the VS network has axonal gap junctions that are critical for optimal prediction. During evasive maneuvers, a VS subpopulation that directly innervates the neck motor center can convey predictive information about the fly's future ego-rotation, potentially crucial for ongoing flight control. These results suggest a novel sensory-motor pathway that links sensory prediction to behavior.

Scientific computational models are crucial for analyzing and understanding complex real-life systems that are otherwise difficult for experimentation. However, the complex behavior and the vast input-output space of these models often make them opaque, slowing the discovery of novel phenomena. In this work, we present HINT (Hessian INTerestingness) -- a new algorithm that can automatically and systematically explore black-box models and highlight local nonlinear interactions in the input-output space of the model. This tool aims to facilitate the discovery of interesting model behaviors that are unknown to the researchers. Using this simple yet powerful tool, we were able to correctly rank all pairwise interactions in known benchmark models and do so faster and with greater accuracy than state-of-the-art methods. We further applied HINT to existing computational neuroscience models, and were able to reproduce important scientific discoveries that were published years after the creation of those models. Finally, we ran HINT on two real-world models (in neuroscience and earth science) and found new behaviors of the model that were of value to domain experts.

Understanding the computational implications of specific synaptic connectivity patterns is a fundamental goal in neuroscience. In particular, the computational role of ubiquitous electrical synapses operating via gap junctions remains elusive. In the fly visual system, the cells in the vertical-system network, which play a key role in visual processing, primarily connect to each other via axonal gap junctions. This network therefore provides a unique opportunity to explore the functional role of gap junctions in sensory information processing. Our information theoretical analysis of a realistic VS network model shows that within 10 ms following the onset of the visual input, the presence of axonal gap junctions enables the VS system to efficiently encode the axis of rotation, θ, of the fly's ego motion. This encoding efficiency, measured in bits, is near-optimal with respect to the physical limits of performance determined by the statistical structure of the visual input itself. The VS network is known to be connected to downstream pathways via a subset of triplets of the vertical system cells; we found that because of the axonal gap junctions, the efficiency of this subpopulation in encoding θ is superior to that of the whole vertical system network and is robust to a wide range of signal to noise ratios. We further demonstrate that this efficient encoding of motion by this subpopulation is necessary for the fly's visually guided behavior, such as banked turns in evasive maneuvers. Because gap junctions are formed among the axons of the vertical system cells, they only impact the system's readout, while maintaining the dendritic input intact, suggesting that the computational principles implemented by neural circuitries may be much richer than previously appreciated based on point neuron models. Our study provides new insights as to how specific network connectivity leads to efficient encoding of sensory stimuli.

An important task performed by a neuron is the selection of relevant inputs from among thousands of synapses impinging on the dendritic tree. Synaptic plasticity enables this by strenghtening a subset of synapses that are, presumably, functionally relevant to the neuron. A different selection mechanism exploits the resonance of the dendritic membranes to preferentially filter synaptic inputs based on their temporal rates. A widely held view is that a neuron has one resonant frequency and thus can pass through one rate. Here we demonstrate through mathematical analyses and numerical simulations that dendritic resonance is inevitably a spatially distributed property; and therefore the resonance frequency varies along the dendrites, and thus endows neurons with a powerful spatiotemporal selection mechanism that is sensitive both to the dendritic location and the temporal structure of the incoming synaptic inputs.

It is commonly accepted that the Inferior Olive (IO) provides a timing signal to the cerebellum. Stable subthreshold oscillations in the IO can facilitate accurate timing by phase-locking spikes to the peaks of the oscillation. Several theoretical models accounting for the synchronized subthreshold oscillations have been proposed, however, two experimental observations remain an enigma. The first is the observation of frequent alterations in the frequency of the oscillations. The second is the observation of constant phase differences between simultaneously recorded neurons. In order to account for these two observations we constructed a canonical network model based on anatomical and physiological data from the IO. The constructed network is characterized by clustering of neurons with similar conductance densities, and by electrical coupling between neurons. Neurons inside a cluster are densely connected with weak strengths, while neurons belonging to different clusters are sparsely connected with stronger connections. We found that this type of network can robustly display stable subthreshold oscillations. The overall frequency of the network changes with the strength of the inter-cluster connections, and phase differences occur between neurons of different clusters. Moreover, the phase differences provide a mechanistic explanation for the experimentally observed propagating waves of activity in the IO. We conclude that the architecture of the network of electrically coupled neurons in combination with modulation of the inter-cluster coupling strengths can account for the experimentally observed frequency changes and the phase differences.

The rich dynamical nature of neurons poses major conceptual and technical challenges for unraveling their nonlinear membrane properties. Traditionally, various current waveforms have been injected at the soma to probe neuron dynamics, but the rationale for selecting specific stimuli has never been rigorously justified. The present experimental and theoretical study proposes a novel framework, inspired by learning theory, for objectively selecting the stimuli that best unravel the neuron's dynamics. The efficacy of stimuli is assessed in terms of their ability to constrain the parameter space of biophysically detailed conductance-based models that faithfully replicate the neuron's dynamics as attested by their ability to generalize well to the neuron's response to novel experimental stimuli. We used this framework to evaluate a variety of stimuli in different types of cortical neurons, ages and animals. Despite their simplicity, a set of stimuli consisting of step and ramp current pulses outperforms synaptic-like noisy stimuli in revealing the dynamics of these neurons. The general framework that we propose paves a new way for defining, evaluating and standardizing effective electrical probing of neurons and will thus lay the foundation for a much deeper understanding of the electrical nature of these highly sophisticated and non-linear devices and of the neuronal networks that they compose.

The thick-tufted layer 5b pyramidal cell extends its dendritic tree to all six layers of the mammalian neocortex and serves as a major building block for the cortical column. L5b pyramidal cells have been the subject of extensive experimental and modeling studies, yet conductance-based models of these cells that faithfully reproduce both their perisomatic Na(+)-spiking behavior as well as key dendritic active properties, including Ca(2+) spikes and back-propagating action potentials, are still lacking. Based on a large body of experimental recordings from both the soma and dendrites of L5b pyramidal cells in adult rats, we characterized key features of the somatic and dendritic firing and quantified their statistics. We used these features to constrain the density of a set of ion channels over the soma and dendritic surface via multi-objective optimization with an evolutionary algorithm, thus generating a set of detailed conductance-based models that faithfully replicate the back-propagating action potential activated Ca(2+) spike firing and the perisomatic firing response to current steps, as well as the experimental variability of the properties. Furthermore, we show a useful way to analyze model parameters with our sets of models, which enabled us to identify some of the mechanisms responsible for the dynamic properties of L5b pyramidal cells as well as mechanisms that are sensitive to morphological changes. This automated framework can be used to develop a database of faithful models for other neuron types. The models we present provide several experimentally-testable predictions and can serve as a powerful tool for theoretical investigations of the contribution of single-cell dynamics to network activity and its computational capabilities.

The firing reliability and precision of an isopotential membrane patch consisting of a realistically large number of ion channels is investigated using a stochastic Hodgkin-Huxley (HH) model. In sharp contrast to the deterministic HH model, the biophysically inspired stochastic model reproduces qualitatively the different reliability and precision characteristics of spike firing in response to DC and fluctuating current input in neocortical neurons, as reported by Mainen & Sejnowski (1995). For DC inputs, spike timing is highly unreliable; the reliability and precision are significantly increased for fluctuating current input. This behavior is critically determined by the relatively small number of excitable channels that are opened near threshold for spike firing rather than by the total number of channels that exist in the membrane patch. Channel fluctuations, together with the inherent bistability in the HH equations, give rise to three additional experimentally observed phenomena: subthreshold oscillations in the membrane voltage for DC input, "spontaneous" spikes for subthreshold inputs, and "missing" spikes for suprathreshold inputs. We suggest that the noise inherent in the operation of ion channels enables neurons to act as "smart" encoders. Slowly varying, uncorrelated inputs are coded with low reliability and accuracy and, hence, the information about such inputs is encoded almost exclusively by the spike rate. On the other hand, correlated presynaptic activity produces sharp fluctuations in the input to the postsynaptic cell, which are then encoded with high reliability and accuracy. In this case, information about the input exists in the exact timing of the spikes. We conclude that channel stochasticity should be considered in realistic models of neurons.

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.

Neurons in the mammalian CNS receive 104-105 synaptic inputs onto their dendritic tree. Each of these inputs may fire spontaneously at a rate of a few spikes per second. Consequently, the cell is bombarded by several hundred synapses in each and every millisecond. An extreme example is the cerebellar Purkinje cell (PC) receiving approximately 100,000 excitatory synapses from the parallel fibers (p.f.s) onto dendritic spines covering the thin dendritic branchlets. What is the effect of the p.f.s activity on the integrative capabilities of the PC? This question is explored theoretically using analytical cable models as well as compartmental models of a morphologically and physiologically characterized PC from the guinea pig cerebellum. The input of individual p.f.s was modeled as a transient conductance change, peaking at 0.4 nS with a rise time of 0.3 msec and a reversal potential of +60 mV relative to rest. We found that already at a firing frequency of a few spikes per second the membrane conductance is several times larger than the membrane conductance in the absence of synaptic activity. As a result, the cable properties of the PC significantly change; the most sensitive parameters are the system time constant (τ0) and the steady-state attenuation factor from dendritic terminal to soma. The implication is that the cable properties of central neurons in freely behaving animals are different from those measured in slice preparation or in anesthetized animals, where most of the synaptic inputs are inactive. We conclude that, because of the large conductance increase produced by the background activity of the p.f.s, the activity of the PC will be altered from this background level either when the p.f.s change their firing frequency for a period of several tens of milliseconds or when a large population of the p.f.s fires during a narrow time window.