Oscillations and synchrony in large-scale cortical network models

Emergent effects of synaptic connectivity on the dynamics of global and local slow waves in a large-scale thalamocortical network model of the human brain

Slow-wave sleep (SWS), characterized by slow oscillations (SOs, <1Hz) of alternating active and silent states in the thalamocortical network, is a primary brain state during Non-Rapid Eye Movement (NREM) sleep. In the last two decades, the traditional view of SWS as a global and uniform whole-brain state has been challenged by a growing body of evidence indicating that SO can be local and can coexist with wake-like activity. However, the mechanisms by which global and local SOs arise from micro-scale neuronal dynamics and network connectivity remain poorly understood. We developed a multi-scale, biophysically realistic human whole-brain thalamocortical network model capable of transitioning between the awake state and SWS, and we investigated the role of connectivity in the spatio-temporal dynamics of sleep SO. We found that the overall strength and a relative balance between long and short-range synaptic connections determined the network state. Importantly, for a range of synaptic strengths, the model demonstrated complex mixed SO states, where periods of synchronized global slow-wave activity were intermittent with the periods of asynchronous local slow-waves. An increase in the overall synaptic strength led to synchronized global SO, while a decrease in synaptic connectivity produced only local slow-waves that would not propagate beyond local areas. These results were compared to human data to validate probable models of biophysically realistic SO. The model producing mixed states provided the best match to the spatial coherence profile and the functional connectivity estimated from human subjects. These findings shed light on how the spatio-temporal properties of SO emerge from local and global cortical connectivity and provide a framework for further exploring the mechanisms and functions of SWS in health and disease.

Coherent olfactory bulb gamma oscillations arise from coupling independent columnar oscillators

Spike timing-based representations of sensory information depend on embedded dynamical frameworks within neuronal networks that establish the rules of local computation and interareal communication. Here, we investigated the dynamical properties of olfactory bulb circuitry in mice of both sexes using microelectrode array recordings from slice and in vivo preparations. Neurochemical activation or optogenetic stimulation of sensory afferents evoked persistent gamma oscillations in the local field potential. These oscillations arose from slower, GABA(A) receptor-independent intracolumnar oscillators coupled by GABA(A)-ergic synapses into a faster, broadly coherent network oscillation. Consistent with the theoretical properties of coupled-oscillator networks, the spatial extent of zero-phase coherence was bounded in slices by the reduced density of lateral interactions. The intact in vivo network, however, exhibited long-range lateral interactions that suffice in simulation to enable zero-phase gamma coherence across the olfactory bulb. The timing of action potentials in a subset of principal neurons was phase-constrained with respect to evoked gamma oscillations. Coupled-oscillator dynamics in olfactory bulb thereby enable a common clock, robust to biological heterogeneities, that is capable of supporting gamma-band spike synchronization and phase coding across the ensemble of activated principal neurons.NEW & NOTEWORTHY Odor stimulation evokes rhythmic gamma oscillations in the field potential of the olfactory bulb, but the dynamical mechanisms governing these oscillations have remained unclear. Establishing these mechanisms is important as they determine the biophysical capacities of the bulbar circuit to, for example, maintain zero-phase coherence across a spatially extended network, or coordinate the timing of action potentials in principal neurons. These properties in turn constrain and suggest hypotheses of sensory coding.

Olfactory receptor neurons generate multiple response motifs, increasing coding space dimensionality

Odorants binding to olfactory receptor neurons (ORNs) trigger bursts of action potentials, providing the brain with its only experience of the olfactory environment. Our recordings made in vivo from locust ORNs showed that odor-elicited firing patterns comprise four distinct response motifs, each defined by a reliable temporal profile. Different odorants could elicit different response motifs from a given ORN, a property we term motif switching. Further, each motif undergoes its own form of sensory adaptation when activated by repeated plume-like odor pulses. A computational model constrained by our recordings revealed that organizing responses into multiple motifs provides substantial benefits for classifying odors and processing complex odor plumes: each motif contributes uniquely to encode the plume's composition and structure. Multiple motifs and motif switching further improve odor classification by expanding coding dimensionality. Our model demonstrated that these response features could provide benefits for olfactory navigation, including determining the distance to an odor source.

Sleep prevents catastrophic forgetting in spiking neural networks by forming a joint synaptic weight representation

Artificial neural networks overwrite previously learned tasks when trained sequentially, a phenomenon known as catastrophic forgetting. In contrast, the brain learns continuously, and typically learns best when new training is interleaved with periods of sleep for memory consolidation. Here we used spiking network to study mechanisms behind catastrophic forgetting and the role of sleep in preventing it. The network could be trained to learn a complex foraging task but exhibited catastrophic forgetting when trained sequentially on different tasks. In synaptic weight space, new task training moved the synaptic weight configuration away from the manifold representing old task leading to forgetting. Interleaving new task training with periods of off-line reactivation, mimicking biological sleep, mitigated catastrophic forgetting by constraining the network synaptic weight state to the previously learned manifold, while allowing the weight configuration to converge towards the intersection of the manifolds representing old and new tasks. The study reveals a possible strategy of synaptic weights dynamics the brain applies during sleep to prevent forgetting and optimize learning.

Optimality of sparse olfactory representations is not affected by network plasticity

The neural representation of a stimulus is repeatedly transformed as it moves from the sensory periphery to deeper layers of the nervous system. Sparsening transformations are thought to increase the separation between similar representations, encode stimuli with great specificity, maximize storage capacity of associative memories, and provide an energy efficient instantiation of information in neural circuits. In the insect olfactory system, odors are initially represented in the periphery as a combinatorial code with relatively simple temporal dynamics. Subsequently, in the antennal lobe this representation is transformed into a dense and complex spatiotemporal activity pattern. Next, in the mushroom body Kenyon cells (KCs), the representation is dramatically sparsened. Finally, in mushroom body output neurons (MBONs), the representation takes on a new dense spatiotemporal format. Here, we develop a computational model to simulate this chain of olfactory processing from the receptor neurons to MBONs. We demonstrate that representations of similar odorants are maximally separated, measured by the distance between the corresponding MBON activity vectors, when KC responses are sparse. Sparseness is maintained across variations in odor concentration by adjusting the feedback inhibition that KCs receive from an inhibitory neuron, the Giant GABAergic neuron. Different odor concentrations require different strength and timing of feedback inhibition for optimal processing. Importantly, as observed in vivo, the KC–MBON synapse is highly plastic, and, therefore, changes in synaptic strength after learning can change the balance of excitation and inhibition, potentially leading to changes in the distance between MBON activity vectors of two odorants for the same level of KC population sparseness. Thus, what is an optimal degree of sparseness before odor learning, could be rendered sub–optimal post learning. Here, we show, however, that synaptic weight changes caused by spike timing dependent plasticity increase the distance between the odor representations from the perspective of MBONs. A level of sparseness that was optimal before learning remains optimal post-learning.

Kenyon cells (KCs) of the mushroom body represent odors as a sparse code. When viewed from the perspective of follower neurons, mushroom body output neurons (MBONs) reveal an optimal level of coding sparseness that maximally separates the representations of odors. However, the KC–MBON synapse is highly plastic and may be potentiated or depressed by odor–driven experience that could, in turn, disrupt the optimality formed by pre–synaptic circuits. Contrary to this expectation, we show that synaptic plasticity based on spike timing of pre- and postsynaptic neurons improves the ability of the system to distinguish between the representations of similar odors while preserving the optimality determined by pre–synaptic circuits.

Simulating human sleep spindle MEG and EEG from ion channel and circuit level dynamics

BACKGROUND

Although they form a unitary phenomenon, the relationship between extracranial M/EEG and transmembrane ion flows is understood only as a general principle rather than as a well-articulated and quantified causal chain.

METHOD

We present an integrated multiscale model, consisting of a neural simulation of thalamus and cortex during stage N2 sleep and a biophysical model projecting cortical current densities to M/EEG fields. Sleep spindles were generated through the interactions of local and distant network connections and intrinsic currents within thalamocortical circuits. 32,652 cortical neurons were mapped onto the cortical surface reconstructed from subjects' MRI, interconnected based on geodesic distances, and scaled-up to current dipole densities based on laminar recordings in humans. MRIs were used to generate a quasi-static electromagnetic model enabling simulated cortical activity to be projected to the M/EEG sensors.

RESULTS

The simulated M/EEG spindles were similar in amplitude and topography to empirical examples in the same subjects. Simulated spindles with more core-dominant activity were more MEG weighted.

COMPARISON WITH EXISTING METHODS

Previous models lacked either spindle-generating thalamic neural dynamics or whole head biophysical modeling; the framework presented here is the first to simultaneously capture these disparate scales.

CONCLUSIONS

This multiscale model provides a platform for the principled quantitative integration of existing information relevant to the generation of sleep spindles, and allows the implications of future findings to be explored. It provides a proof of principle for a methodological framework allowing large-scale integrative brain oscillations to be understood in terms of their underlying channels and synapses.

Highly Compact Artificial Memristive Neuron with Low Energy Consumption

Neuromorphic systems aim to implement large-scale artificial neural network on hardware to ultimately realize human-level intelligence. The recent development of nonsilicon nanodevices has opened the huge potential of full memristive neural networks (FMNN), consisting of memristive neurons and synapses, for neuromorphic applications. Unlike the widely reported memristive synapses, the development of artificial neurons on memristive devices has less progress. Sophisticated neural dynamics is the major obstacle behind the lagging. Here a rich dynamics-driven artificial neuron is demonstrated, which successfully emulates partial essential neural features of neural processing, including leaky integration, automatic threshold-driven fire, and self-recovery, in a unified manner. The realization of bioplausible artificial neurons on a single device with ultralow power consumption paves the way for constructing energy-efficient large-scale FMNN and may boost the development of neuromorphic systems with high density, low power, and fast speed.

New class of reduced computationally efficient neuronal models for large-scale simulations of brain dynamics

During slow-wave sleep, brain electrical activity is dominated by the slow (< 1 Hz) electroencephalogram (EEG) oscillations characterized by the periodic transitions between active (or Up) and silent (or Down) states in the membrane voltage of the cortical and thalamic neurons. Sleep slow oscillation is believed to play critical role in consolidation of recent memories. Past computational studies, based on the Hodgkin-Huxley type neuronal models, revealed possible intracellular and network mechanisms of the neuronal activity during sleep, however, they failed to explore the large-scale cortical network dynamics depending on collective behavior in the large populations of neurons. In this new study, we developed a novel class of reduced discrete time spiking neuron models for large-scale network simulations of wake and sleep dynamics. In addition to the spiking mechanism, the new model implemented nonlinearities capturing effects of the leak current, the Ca²⁺ dependent K⁺ current and the persistent Na⁺ current that were found to be critical for transitions between Up and Down states of the slow oscillation. We applied the new model to study large-scale two-dimensional cortical network activity during slow-wave sleep. Our study explained traveling wave dynamics and characteristic synchronization properties of transitions between Up and Down states of the slow oscillation as observed in vivo in recordings from cats. We further predict a critical role of synaptic noise and slow adaptive currents for spike sequence replay as found during sleep related memory consolidation.

A coupled-oscillator model of olfactory bulb gamma oscillations

The olfactory bulb transforms not only the information content of the primary sensory representation, but also its underlying coding metric. High-variance, slow-timescale primary odor representations are transformed by bulbar circuitry into secondary representations based on principal neuron spike patterns that are tightly regulated in time. This emergent fast timescale for signaling is reflected in gamma-band local field potentials, presumably serving to efficiently integrate olfactory sensory information into the temporally regulated information networks of the central nervous system. To understand this transformation and its integration with interareal coordination mechanisms requires that we understand its fundamental dynamical principles. Using a biophysically explicit, multiscale model of olfactory bulb circuitry, we here demonstrate that an inhibition-coupled intrinsic oscillator framework, pyramidal resonance interneuron network gamma (PRING), best captures the diversity of physiological properties exhibited by the olfactory bulb. Most importantly, these properties include global zero-phase synchronization in the gamma band, the phase-restriction of informative spikes in principal neurons with respect to this common clock, and the robustness of this synchronous oscillatory regime to multiple challenging conditions observed in the biological system. These conditions include substantial heterogeneities in afferent activation levels and excitatory synaptic weights, high levels of uncorrelated background activity among principal neurons, and spike frequencies in both principal neurons and interneurons that are irregular in time and much lower than the gamma frequency. This coupled cellular oscillator architecture permits stable and replicable ensemble responses to diverse sensory stimuli under various external conditions as well as to changes in network parameters arising from learning-dependent synaptic plasticity.

Multi-layer network utilizing rewarded spike time dependent plasticity to learn a foraging task

Neural networks with a single plastic layer employing reward modulated spike time dependent plasticity (STDP) are capable of learning simple foraging tasks. Here we demonstrate advanced pattern discrimination and continuous learning in a network of spiking neurons with multiple plastic layers. The network utilized both reward modulated and non-reward modulated STDP and implemented multiple mechanisms for homeostatic regulation of synaptic efficacy, including heterosynaptic plasticity, gain control, output balancing, activity normalization of rewarded STDP and hard limits on synaptic strength. We found that addition of a hidden layer of neurons employing non-rewarded STDP created neurons that responded to the specific combinations of inputs and thus performed basic classification of the input patterns. When combined with a following layer of neurons implementing rewarded STDP, the network was able to learn, despite the absence of labeled training data, discrimination between rewarding patterns and the patterns designated as punishing. Synaptic noise allowed for trial-and-error learning that helped to identify the goal-oriented strategies which were effective in task solving. The study predicts a critical set of properties of the spiking neuronal network with STDP that was sufficient to solve a complex foraging task involving pattern classification and decision making.

This study explores how intelligent behavior emerges from the basic principles known at the cellular level of biological neuronal network dynamics. Compared to the approaches used in the artificial intelligence community, we applied biologically realistic modeling of neuronal dynamics and plasticity. The building blocks of the model are spiking neurons, spike-time dependent plasticity (STDP) and homeostatic rules, known experimentally, which are shown to play a fundamental role in both keeping the network stable and capable of continous learning. Our study predicts that a combination of these principles makes possible a foraging behavior in a previously unknown environment, including pattern classification to distinct between environment shapes which are rewarded and those which are punished and decision making to select the optimal strategy to acquire the maximal number of the rewarded elements. To solve this complex task we used multi-layer neuronal processing that implemented pattern generalization by unsupervised STDP at the earlier processing step, as commonly observed in the animal and human sensory processing, followed by reinforcement learning at the later steps. In the model, the intelligent behavior emerged spontaneously due to the network organization implementing both local unsupervised plasticity and reward feedback resulting from a successful behavior in the environment.

Collective behavior of large-scale neural networks with GPU acceleration

In this paper, the collective behaviors of a small-world neuronal network motivated by the anatomy of a mammalian cortex based on both Izhikevich model and Rulkov model are studied. The Izhikevich model can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Rulkov model is in the form of difference equations that generate a sequence of membrane potential samples in discrete moments of time to improve computational efficiency. These two models are suitable for the construction of large scale neural networks. By varying some key parameters, such as the connection probability and the number of nearest neighbor of each node, the coupled neurons will exhibit types of temporal and spatial characteristics. It is demonstrated that the implementation of GPU can achieve more and more acceleration than CPU with the increasing of neuron number and iterations. These two small-world network models and GPU acceleration give us a new opportunity to reproduce the real biological network containing a large number of neurons.

Biophysical constraints on lateral inhibition in the olfactory bulb

The mitral cells (MCs) of the mammalian olfactory bulb (OB) constitute one of two populations of principal neurons (along with middle/deep tufted cells) that integrate afferent olfactory information with top-down inputs and intrinsic learning and deliver output to downstream olfactory areas. MC activity is regulated in part by inhibition from granule cells, which form reciprocal synapses with MCs along the extents of their lateral dendrites. However, with MC lateral dendrites reaching over 1.5 mm in length in rats, the roles of distal inhibitory synapses pose a quandary. Here, we systematically vary the properties of a MC model to assess the capacity of inhibitory synaptic inputs on lateral dendrites to influence afferent information flow through MCs. Simulations using passivized models with varying dendritic morphologies and synaptic properties demonstrated that, even with unrealistically favorable parameters, passive propagation fails to convey effective inhibitory signals to the soma from distal sources. Additional simulations using an active model exhibiting action potentials, subthreshold oscillations, and a dendritic morphology closely matched to experimental values further confirmed that distal synaptic inputs along the lateral dendrite could not exert physiologically relevant effects on MC spike timing at the soma. Larger synaptic conductances representative of multiple simultaneous inputs were not sufficient to compensate for the decline in signal with distance. Reciprocal synapses on distal MC lateral dendrites may instead serve to maintain a common fast oscillatory clock across the OB by delaying spike propagation within the lateral dendrites themselves.

Classification of odorants across layers in locust olfactory pathway

Olfactory processing takes place across multiple layers of neurons from the transduction of odorants in the periphery, to odor quality processing, learning, and decision making in higher olfactory structures. In insects, projection neurons (PNs) in the antennal lobe send odor information to the Kenyon cells (KCs) of the mushroom bodies and lateral horn neurons (LHNs). To examine the odor information content in different structures of the insect brain, antennal lobe, mushroom bodies and lateral horn, we designed a model of the olfactory network based on electrophysiological recordings made in vivo in the locust. We found that populations of all types (PNs, LHNs, and KCs) had lower odor classification error rates than individual cells of any given type. This improvement was quantitatively different from that observed using uniform populations of identical neurons compared with spatially structured population of neurons tuned to different odor features. This result, therefore, reflects an emergent network property. Odor classification improved with increasing stimulus duration: for similar odorants, KC and LHN ensembles reached optimal discrimination within the first 300-500 ms of the odor response. Performance improvement with time was much greater for a population of cells than for individual neurons. We conclude that, for PNs, LHNs, and KCs, ensemble responses are always much more informative than single-cell responses, despite the accumulation of noise along with odor information.

Feed-Forward versus Feedback Inhibition in a Basic Olfactory Circuit

Inhibitory interneurons play critical roles in shaping the firing patterns of principal neurons in many brain systems. Despite difference in the anatomy or functions of neuronal circuits containing inhibition, two basic motifs repeatedly emerge: feed-forward and feedback. In the locust, it was proposed that a subset of lateral horn interneurons (LHNs), provide feed-forward inhibition onto Kenyon cells (KCs) to maintain their sparse firing—a property critical for olfactory learning and memory. But recently it was established that a single inhibitory cell, the giant GABAergic neuron (GGN), is the main and perhaps sole source of inhibition in the mushroom body, and that inhibition from this cell is mediated by a feedback (FB) loop including KCs and the GGN. To clarify basic differences in the effects of feedback vs. feed-forward inhibition in circuit dynamics we here use a model of the locust olfactory system. We found both inhibitory motifs were able to maintain sparse KCs responses and provide optimal odor discrimination. However, we further found that only FB inhibition could create a phase response consistent with data recorded in vivo. These findings describe general rules for feed-forward versus feedback inhibition and suggest GGN is potentially capable of providing the primary source of inhibition to the KCs. A better understanding of how inhibitory motifs impact post-synaptic neuronal activity could be used to reveal unknown inhibitory structures within biological networks.

Understanding how inhibitory neurons interact with excitatory neurons is critical for understanding the behaviors of neuronal networks. Here we address this question with simple but biologically relevant models based on the anatomy of the locust olfactory pathway. Two ubiquitous and basic inhibitory motifs were tested: feed-forward and feedback. Feed-forward inhibition typically occurs between different brain areas when excitatory neurons excite inhibitory cells, which then inhibit a group of postsynaptic excitatory neurons outside of the initializing excitatory neurons’ area. On the other hand, the feedback inhibitory motif requires a population of excitatory neurons to drive the inhibitory cells, which in turn inhibit the same population of excitatory cells. We found the type of the inhibitory motif determined the timing with which each group of cells fired action potentials in comparison to one another (relative timing). It also affected the range of inhibitory neurons’ activity, with the inhibitory neurons having a wider range in the feedback circuit than that in the feed-forward one. These results will allow predicting the type of the connectivity structure within unexplored biological circuits given only electrophysiological recordings.

A spiking network model of decision making employing rewarded STDP

Reward-modulated spike timing dependent plasticity (STDP) combines unsupervised STDP with a reinforcement signal that modulates synaptic changes. It was proposed as a learning rule capable of solving the distal reward problem in reinforcement learning. Nonetheless, performance and limitations of this learning mechanism have yet to be tested for its ability to solve biological problems. In our work, rewarded STDP was implemented to model foraging behavior in a simulated environment. Over the course of training the network of spiking neurons developed the capability of producing highly successful decision-making. The network performance remained stable even after significant perturbations of synaptic structure. Rewarded STDP alone was insufficient to learn effective decision making due to the difficulty maintaining homeostatic equilibrium of synaptic weights and the development of local performance maxima. Our study predicts that successful learning requires stabilizing mechanisms that allow neurons to balance their input and output synapses as well as synaptic noise.

Timing control by redundant inhibitory neuronal circuits

Rhythms and timing control of sequential activity in the brain is fundamental to cognition and behavior. Although experimental and theoretical studies support the understanding that neuronal circuits are intrinsically capable of generating different time intervals, the dynamical origin of the phenomenon of functionally dependent timing control is still unclear. Here, we consider a new mechanism that is related to the multi-neuronal cooperative dynamics in inhibitory brain motifs consisting of a few clusters. It is shown that redundancy and diversity of neurons within each cluster enhances the sensitivity of the timing control with the level of neuronal excitation of the whole network. The generality of the mechanism is shown to work on two different neuronal models: a conductance-based model and a map-based model.

Oscillations and synchrony in a cortical neural network

In this paper, the oscillations and synchronization status of two different network connectivity patterns based on Izhikevich model are studied. One of the connectivity patterns is a randomly connected neuronal network, the other one is a small-world neuronal network. This Izhikevich model is a simple model which can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Detailed investigations reveal that by varying some key parameters, such as the connection weights of neurons, the external current injection, the noise of intensity and the neuron number, this neuronal network will exhibit various collective behaviors in randomly coupled neuronal network. In addition, we show that by changing the number of nearest neighbor and connection probability in small-world topology can also affect the collective dynamics of neuronal activity. These results may be instructive in understanding the collective dynamics of mammalian cortex.

Hybrid discrete-time neural networks

Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast-slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.

Robust observer-based tracking control of hodgkin-huxley neuron systems under environmental disturbances

A nervous system consists of a large number of highly interconnected nerve cells. Nerve cells communicate by generation and transmission of short electrical pulses (action potential). In addition, membrane voltage is the only measurable state in nervous systems. A robust observer-based model reference tracking control is proposed for Hodgkin-Huxley (HH) neuron systems to generate a desired reference response in spite of environmental noises, uncertain initial values, and diffusion currents from other interconnected nerve cells. In order to simplify the robust tracking control design of nonlinear stochastic HH neuron systems, a fuzzy interpolation method is employed to interpolate several linear stochastic systems to approximate a nonlinear stochastic HH neuron system so that the nonlinear robust tracking control problem can be solved by the linear matrix inequality (LMI) technique with the help of Robust Control Toolbox in Matlab. The proposed robust observer-based tracking control scheme can provide new methods for desired action potential generation, suppression of oscillations, and blockage of action potential transmission under environmental noise and diffusion currents. These new methods are useful for patients with different neuron system dysfunctions. Finally, three simulation examples of tracking control of nervous systems are given to illustrate the design procedure and confirm the tracking performance of the proposed method.

Information-geometric measure of 3-neuron firing patterns characterizes scale-dependence in cortical networks

To understand the functional connectivity of neural networks, it is important to develop simple and incisive descriptors of multineuronal firing patterns. Analysis at the pairwise level has proven to be a powerful approach in the retina, but it may not suffice to understand complex cortical networks. Here we address the problem of describing interactions among triplets of neurons. We consider two approaches: an information-geometric measure (Amari 2001), which we call the "strain," and the Kullback-Leibler divergence. While both approaches can be used to assess whether firing patterns differ from those predicted by a pairwise maximum-entropy model, the strain provides additional information. Specifically, when the observed firing patterns differ from those predicted by a pairwise model, the strain indicates the nature of this difference--whether there is an excess or a deficit of synchrony--while the Kullback-Leibler divergence only indicates the magnitude of the difference. We show that the strain has technical advantages, including ease of calculation of confidence bounds and bias, and robustness to the kinds of spike-sorting errors associated with tetrode recordings. We demonstrate the biological importance of these points via an analysis of multineuronal firing patterns in primary visual cortex. There is a striking scale-dependent behavior of triplet firing patterns: deviations from the pairwise model are substantial when the neurons are within 300 microns of each other, and negligible when they are at a distance of >600 microns. The strain identifies a consistent pattern to these interactions: when triplet interactions are present, the strain is nearly always negative, indicating that there is less synchrony than would be expected from the pairwise interactions alone.

Frequency transitions in odor-evoked neural oscillations

In many species, sensory stimuli elicit the oscillatory synchronization of groups of neurons. What determines the properties of these oscillations? In the olfactory system of the moth, we found that odors elicited oscillatory synchronization through a neural mechanism like that described in locust and Drosophila. During responses to long odor pulses, oscillations suddenly slowed as net olfactory receptor neuron (ORN) output decreased; thus, stimulus intensity appeared to determine oscillation frequency. However, changing the concentration of the odor had little effect upon oscillatory frequency. Our recordings in vivo and computational models based on these results suggested that the main effect of increasing odor concentration was to recruit additional, less well-tuned ORNs whose firing rates were tightly constrained by adaptation and saturation. Thus, in the periphery, concentration is encoded mainly by the size of the responsive ORN population, and oscillation frequency is set by the adaptation and saturation of this response.