Mean-field analysis of synaptic alterations underlying deficient cortical gamma oscillations in schizophrenia.

Deficient gamma oscillations in the prefrontal cortex (PFC) of individuals with schizophrenia (SZ) are proposed to arise from alterations in the excitatory drive to fast-spiking interneurons (E→I) and in the inhibitory drive from these interneurons to excitatory neurons (I→E). Consistent with this idea, prior postmortem studies showed lower levels of molecular and structural markers for the strength of E→I and I→E synapses and also greater variability in E→I synaptic strength in PFC of SZ. Moreover, simulating these alterations in a network of quadratic integrate-and-fire (QIF) neurons revealed a synergistic effect of their interactions on reducing gamma power. In this study, we aimed to investigate the dynamical nature of this synergistic interaction at macroscopic level by deriving a mean-field description of the QIF model network that consists of all-to-all connected excitatory neurons and fast-spiking interneurons. Through a series of numerical simulations and bifurcation analyses, findings from our mean-field model showed that the macroscopic dynamics of gamma oscillations are synergistically disrupted by the interactions among lower strength of E→I and I→E synapses and greater variability in E→I synaptic strength. Furthermore, the two-dimensional bifurcation analyses showed that this synergistic interaction is primarily driven by the shift in Hopf bifurcation due to lower E→I synaptic strength. Together, these simulations predict the nature of dynamical mechanisms by which multiple synaptic alterations interact to robustly reduce PFC gamma power in SZ, and highlight the utility of mean-field model to study macroscopic neural dynamics and their alterations in the illness.

Altered Rbfox1-Vamp1 pathway and prefrontal cortical dysfunction in schizophrenia.

Deficient gamma oscillations in prefrontal cortex (PFC) of individuals with schizophrenia appear to involve impaired inhibitory drive from parvalbumin-expressing interneurons (PVIs). Inhibitory drive from PVIs is regulated, in part, by RNA binding fox-1 homolog 1 (Rbfox1). Rbfox1 is spliced into nuclear or cytoplasmic isoforms, which regulate alternative splicing or stability of their target transcripts, respectively. One major target of cytoplasmic Rbfox1 is vesicle associated membrane protein 1 (Vamp1). Vamp1 mediates GABA release probability from PVIs, and the loss of Rbfox1 reduces Vamp1 levels which in turn impairs cortical inhibition. In this study, we investigated if the Rbfox1-Vamp1 pathway is altered in PVIs in PFC of individuals with schizophrenia by utilizing a novel strategy that combines multi-label in situ hybridization and immunohistochemistry. In the PFC of 20 matched pairs of schizophrenia and comparison subjects, cytoplasmic Rbfox1 protein levels were significantly lower in PVIs in schizophrenia and this deficit was not attributable to potential methodological confounds or schizophrenia-associated co-occurring factors. In a subset of this cohort, Vamp1 mRNA levels in PVIs were also significantly lower in schizophrenia and were predicted by lower cytoplasmic Rbfox1 protein levels across individual PVIs. To investigate the functional impact of Rbfox1-Vamp1 alterations in schizophrenia, we simulated the effect of lower GABA release probability from PVIs on gamma power in a computational model network of pyramidal neurons and PVIs. Our simulations showed that lower GABA release probability reduces gamma power by disrupting network synchrony while minimally affecting network activity. Finally, lower GABA release probability synergistically interacted with lower strength of inhibition from PVIs in schizophrenia to reduce gamma power non-linearly. Together, our findings suggest that the Rbfox1-Vamp1 pathway in PVIs is impaired in schizophrenia and that this alteration likely contributes to deficient PFC gamma power in the illness.

An oscillatory mechanism for multi-level storage in short-term memory.

Oscillatory activity is commonly observed during the maintenance of information in short-term memory, but its role remains unclear. Non-oscillatory models of short-term memory storage are able to encode stimulus identity through their spatial patterns of activity, but are typically limited to either an all-or-none representation of stimulus amplitude or exhibit a biologically implausible exact-tuning condition. Here we demonstrate a simple mechanism by which oscillatory input enables a circuit to generate persistent or sequential activity that encodes information not only in the spatial pattern of activity, but also in the amplitude of activity. This is accomplished through a phase-locking phenomenon that permits many different amplitudes of persistent activity to be stored without requiring exact tuning of model parameters. Altogether, this work proposes a class of models for the storage of information in working memory, a potential role for brain oscillations, and a dynamical mechanism for maintaining multi-stable neural representations.

Cross-scale excitability in networks of quadratic integrate-and-fire neurons.

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. Here, we show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.

BioSimulators: a central registry of simulation engines and services for recommending specific tools.

Computational models have great potential to accelerate bioscience, bioengineering, and medicine. However, it remains challenging to reproduce and reuse simulations, in part, because the numerous formats and methods for simulating various subsystems and scales remain siloed by different software tools. For example, each tool must be executed through a distinct interface. To help investigators find and use simulation tools, we developed BioSimulators (https://biosimulators.org), a central registry of the capabilities of simulation tools and consistent Python, command-line and containerized interfaces to each version of each tool. The foundation of BioSimulators is standards, such as CellML, SBML, SED-ML and the COMBINE archive format, and validation tools for simulation projects and simulation tools that ensure these standards are used consistently. To help modelers find tools for particular projects, we have also used the registry to develop recommendation services. We anticipate that BioSimulators will help modelers exchange, reproduce, and combine simulations.

A multiscale multicellular spatiotemporal model of local influenza infection and immune response.

Respiratory viral infections pose a serious public health concern, from mild seasonal influenza to pandemics like those of SARS-CoV-2. Spatiotemporal dynamics of viral infection impact nearly all aspects of the progression of a viral infection, like the dependence of viral replication rates on the type of cell and pathogen, the strength of the immune response and localization of infection. Mathematical modeling is often used to describe respiratory viral infections and the immune response to them using ordinary differential equation (ODE) models. However, ODE models neglect spatially-resolved biophysical mechanisms like lesion shape and the details of viral transport, and so cannot model spatial effects of a viral infection and immune response. In this work, we develop a multiscale, multicellular spatiotemporal model of influenza infection and immune response by combining non-spatial ODE modeling and spatial, cell-based modeling. We employ cellularization, a recently developed method for generating spatial, cell-based, stochastic models from non-spatial ODE models, to generate much of our model from a calibrated ODE model that describes infection, death and recovery of susceptible cells and innate and adaptive responses during influenza infection, and develop models of cell migration and other mechanisms not explicitly described by the ODE model. We determine new model parameters to generate agreement between the spatial and original ODE models under certain conditions, where simulation replicas using our model serve as microconfigurations of the ODE model, and compare results between the models to investigate the nature of viral exposure and impact of heterogeneous infection on the time-evolution of the viral infection. We found that using spatially homogeneous initial exposure conditions consistently with those employed during calibration of the ODE model generates far less severe infection, and that local exposure to virus must be multiple orders of magnitude greater than a uniformly applied exposure to all available susceptible cells. This strongly suggests a prominent role of localization of exposure in influenza A infection. We propose that the particularities of the microenvironment to which a virus is introduced plays a dominant role in disease onset and progression, and that spatially resolved models like ours may be important to better understand and more reliably predict future health states based on susceptibility of potential lesion sites using spatially resolved patient data of the state of an infection. We can readily integrate the immune response components of our model into other modeling and simulation frameworks of viral infection dynamics that do detailed modeling of other mechanisms like viral internalization and intracellular viral replication dynamics, which are not explicitly represented in the ODE model. We can also combine our model with available experimental data and modeling of exposure scenarios and spatiotemporal aspects of mechanisms like mucociliary clearance that are only implicitly described by the ODE model, which would significantly improve the ability of our model to present spatially resolved predictions about the progression of influenza infection and immune response.

Compartmental Model Suggests Importance of Innate Immune Response to COVID-19 Infection in Rhesus Macaques.

The pandemic outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has quickly spread worldwide, creating a serious health crisis. The virus is primarily associated with flu-like symptoms but can also lead to severe pathologies and death. We here present an ordinary differential equation model of the intrahost immune response to SARS-CoV-2 infection, fitted to experimental data gleaned from rhesus macaques. The model is calibrated to data from a nonlethal infection, but the model can replicate behavior from various lethal scenarios as well. We evaluate the sensitivity of the model to biologically relevant parameters governing the strength and efficacy of the immune response. We also simulate the effect of both anti-inflammatory and antiviral drugs on the host immune response and demonstrate the ability of the model to lessen the severity of a formerly lethal infection with the addition of the appropriately calibrated drug. Our model emphasizes the importance of tight control of the innate immune response for host survival and viral clearance.

Kcns3 deficiency disrupts Parvalbumin neuron physiology in mouse prefrontal cortex: Implications for the pathophysiology of schizophrenia.

The unique fast spiking (FS) phenotype of cortical parvalbumin-positive (PV) neurons depends on the expression of multiple subtypes of voltage-gated potassium channels (Kv). PV neurons selectively express Kcns3, the gene encoding Kv9.3 subunits, suggesting that Kcns3 expression is critical for the FS phenotype. KCNS3 expression is lower in PV neurons in the neocortex of subjects with schizophrenia, but the effects of this alteration are unclear, because Kv9.3 subunit function is poorly understood. Therefore, to assess the role of Kv9.3 subunits in PV neuron function, we combined gene expression analyses, computational modeling, and electrophysiology in acute slices from the cortex of Kcns3-deficient mice. Kcns3 mRNA levels were ~ 50% lower in cortical PV neurons from Kcns3-deficient relative to wildtype mice. While silent per se, Kv9.3 subunits are believed to amplify the Kv2.1 current in Kv2.1-Kv9.3 channel complexes. Hence, to assess the consequences of reducing Kv9.3 levels, we simulated the effects of decreasing the Kv2.1-mediated current in a computational model. The FS cell model with reduced Kv2.1 produced spike trains with irregular inter-spike intervals, or stuttering, and greater Na+ channel inactivation. As in the computational model, PV basket cells (PVBCs) from Kcns3-deficient mice displayed spike trains with strong stuttering, which depressed PVBC firing. Moreover, Kcns3 deficiency impaired the recruitment of PVBC firing at gamma frequency by stimuli mimicking synaptic input observed during cortical UP states. Our data indicate that Kv9.3 subunits are critical for PVBC physiology and suggest that KCNS3 deficiency in schizophrenia could impair PV neuron firing, possibly contributing to deficits in cortical gamma oscillations in the illness.

A computational model of the shrimp-goby escape and communication system.

Fish escape from approaching threats via a stereotyped escape behavior. This behavior, and the underlying neural circuit organized around the Mauthner cell command neurons, have both been extensively investigated experimentally, mainly in two laboratory model organisms, the goldfish and the zebrafish. However, fish biodiversity is enormous, a number of variants of the basal escape behavior exist. In marine gobies (a family of small benthic fishes) which share burrows with alpheid shrimp, the escape behavior has likely been partially modified into a tactile communication system which allow the fish to communicate the approach of a predatory fish to the shrimp. In this communication system, the goby responds to intermediate-strength threats with a brief tail-flick which the shrimp senses with its antennae.We investigated the shrimp goby escape and communication system with computational models. We asked how the circuitry of the basal escape behavior could be modified to produce behavior akin to the shrimp-goby communication system. In a simple model, we found that mutual inhibitions between Mauthner cells can be tuned to produce an oscillatory response to intermediate strength inputs, albeit only in a narrow parameter range.Using a more detailed model, we found that two modifications of the fish locomotor system transform it into a model reproducing the shrimp goby behavior. These modifications are: 1. modifying the central pattern generator which drives swimming such that it is quiescent when receiving no inputs; 2. introducing a direct sensory input to this central pattern generator, bypassing the Mauthner cells.

Olfactory navigation in the real world: Simple local search strategies for turbulent environments.

Olfaction informs animal navigation for foraging, social interaction, and threat evasion. However, turbulent flow on the spatial scales of most animal navigation leads to intermittent odor information and presents a challenge to simple gradient-ascent navigation. Here we present two strategies for iterative gradient estimation and navigation via olfactory cues in 2D space: tropotaxis, spatial concentration comparison (i.e., instantaneous comparison between lateral olfactory sensors on a navigating animal) and klinotaxis, spatiotemporal concentration comparison (i.e., comparison between two subsequent concentration samples as the animal moves through space). We then construct a hybrid model that uses klinotaxis but utilizes tropotactic information to guide its spatial sampling strategy. We find that for certain body geometries in which bilateral sensors are closely-spaced (e.g., mammalian nares), klinotaxis outperforms tropotaxis; for widely-spaced sensors (e.g., arthropod antennae), tropotaxis outperforms klinotaxis. We find that both navigation strategies perform well on smooth odor gradients and are robust against noisy gradients represented by stochastic odor models and real turbulent flow data. In some parameter regimes, the hybrid model outperforms klinotaxis alone, but not tropotaxis.

Direction-selective motion discrimination by traveling waves in visual cortex.

The majority of neurons in primary visual cortex respond selectively to bars of light that have a specific orientation and move in a specific direction. The spatial and temporal responses of such neurons are non-separable. How neurons accomplish that computational feat without resort to explicit time delays is unknown. We propose a novel neural mechanism whereby visual cortex computes non-separable responses by generating endogenous traveling waves of neural activity that resonate with the space-time signature of the visual stimulus. The spatiotemporal characteristics of the response are defined by the local topology of excitatory and inhibitory lateral connections in the cortex. We simulated the interaction between endogenous traveling waves and the visual stimulus using spatially distributed populations of excitatory and inhibitory neurons with Wilson-Cowan dynamics and inhibitory-surround coupling. Our model reliably detected visual gratings that moved with a given speed and direction provided that we incorporated neural competition to suppress false motion signals in the opposite direction. The findings suggest that endogenous traveling waves in visual cortex can impart direction-selectivity on neural responses without resort to explicit time delays. They also suggest a functional role for motion opponency in eliminating false motion signals.

Olfactory Navigation and the Receptor Nonlinearity.

The demands on a sensory system depend not only on the statistics of its inputs but also on the task. In olfactory navigation, for example, the task is to find the plume source; allocation of sensory resources may therefore be driven by aspects of the plume that are informative about source location, rather than concentration per se. Here we explore the implications of this idea for encoding odor concentration. To formalize the notion that sensory resources are limited, we considered coding strategies that partitioned the odor concentration range into a set of discriminable intervals. We developed a dynamic programming algorithm that, given the distribution of odor concentrations at several locations, determines the partitioning that conveys the most information about location. We applied this analysis to planar laser-induced fluorescence measurements of spatiotemporal odor fields with realistic advection speeds (5-20 cm/s), with or without a nearby boundary or obstacle. Across all environments, the optimal coding strategy allocated more resources (i.e., more and finer discriminable intervals) to the upper end of the concentration range than would be expected from histogram equalization, the optimal strategy if the goal were to reconstruct the plume, rather than to navigate. Finally, we show that ligand binding, as captured by the Hill equation, transforms odorant concentration into response levels in a way that approximates information maximization for navigation. This behavior occurs when the Hill dissociation constant is near the mean odor concentration, an adaptive set-point that has been observed in the olfactory system of flies.SIGNIFICANCE STATEMENT The first step of olfactory processing is receptor binding, and the resulting relationship between odorant concentration and the bound receptor fraction is a saturating one. While this Hill nonlinearity can be viewed as a distortion that is imposed by the biophysics of receptor binding, here we show that it also plays an important information-processing role in olfactory navigation. Specifically, by combining a novel dynamic-programming algorithm with physical measurements of turbulent plumes, we determine the optimal strategy for encoding odor concentration when the goal is to determine location. This strategy is distinct from histogram equalization, the strategy that maximizes information about plume concentration, and is closely approximated by the Hill nonlinearity when the binding constant is near the ambient mean.

Emergent mechanics of actomyosin drive punctuated contractions and shape network morphology in the cell cortex.

Filamentous actin (F-actin) and non-muscle myosin II motors drive cell motility and cell shape changes that guide large scale tissue movements during embryonic morphogenesis. To gain a better understanding of the role of actomyosin in vivo, we have developed a two-dimensional (2D) computational model to study emergent phenomena of dynamic unbranched actomyosin arrays in the cell cortex. These phenomena include actomyosin punctuated contractions, or "actin asters" that form within quiescent F-actin networks. Punctuated contractions involve both formation of high intensity aster-like structures and disassembly of those same structures. Our 2D model allows us to explore the kinematics of filament polarity sorting, segregation of motors, and morphology of F-actin arrays that emerge as the model structure and biophysical properties are varied. Our model demonstrates the complex, emergent feedback between filament reorganization and motor transport that generate as well as disassemble actin asters. Since intracellular actomyosin dynamics are thought to be controlled by localization of scaffold proteins that bind F-actin or their myosin motors we also apply our 2D model to recapitulate in vitro studies that have revealed complex patterns of actomyosin that assemble from patterning filaments and motor complexes with microcontact printing. Although we use a minimal representation of filament, motor, and cross-linker biophysics, our model establishes a framework for investigating the role of other actin binding proteins, how they might alter actomyosin dynamics, and makes predictions that can be tested experimentally within live cells as well as within in vitro models.

A multiple timescales approach to bridging spiking- and population-level dynamics.

A rigorous bridge between spiking-level and macroscopic quantities is an on-going and well-developed story for asynchronously firing neurons, but focus has shifted to include neural populations exhibiting varying synchronous dynamics. Recent literature has used the Ott-Antonsen ansatz (2008) to great effect, allowing a rigorous derivation of an order parameter for large oscillator populations. The ansatz has been successfully applied using several models including networks of Kuramoto oscillators, theta models, and integrate-and-fire neurons, along with many types of network topologies. In the present study, we take a converse approach: given the mean field dynamics of slow synapses, we predict the synchronization properties of finite neural populations. The slow synapse assumption is amenable to averaging theory and the method of multiple timescales. Our proposed theory applies to two heterogeneous populations of N excitatory n-dimensional and N inhibitory m-dimensional oscillators with homogeneous synaptic weights. We then demonstrate our theory using two examples. In the first example, we take a network of excitatory and inhibitory theta neurons and consider the case with and without heterogeneous inputs. In the second example, we use Traub models with calcium for the excitatory neurons and Wang-Buzsáki models for the inhibitory neurons. We accurately predict phase drift and phase locking in each example even when the slow synapses exhibit non-trivial mean-field dynamics.

Information-theoretic analysis of realistic odor plumes: What cues are useful for determining location?

Many species rely on olfaction to navigate towards food sources or mates. Olfactory navigation is a challenging task since odor environments are typically turbulent. While time-averaged odor concentration varies smoothly with the distance to the source, instaneous concentrations are intermittent and obtaining stable averages takes longer than the typical intervals between animals' navigation decisions. How to effectively sample from the odor distribution to determine sampling location is the focus in this article. To investigate which sampling strategies are most informative about the location of an odor source, we recorded three naturalistic stimuli with planar lased-induced fluorescence and used an information-theoretic approach to quantify the information that different sampling strategies provide about sampling location. Specifically, we compared multiple sampling strategies based on a fixed number of coding bits for encoding the olfactory stimulus. When the coding bits were all allocated to representing odor concentration at a single sensor, information rapidly saturated. Using the same number of coding bits in two sensors provides more information, as does coding multiple samples at different times. When accumulating multiple samples at a fixed location, the temporal sequence does not yield a large amount of information and can be averaged with minimal loss. Furthermore, we show that histogram-equalization is not the most efficient way to use coding bits when using the olfactory sample to determine location.

Relationship between the mechanisms of gamma rhythm generation and the magnitude of the macroscopic phase response function in a population of excitatory and inhibitory modified quadratic integrate-and-fire neurons.

Gamma oscillations are thought to play an important role in brain function. Interneuron gamma (ING) and pyramidal interneuron gamma (PING) mechanisms have been proposed as generation mechanisms for these oscillations. However, the relation between the generation mechanisms and the dynamical properties of the gamma oscillation are still unclear. Among the dynamical properties of the gamma oscillation, the phase response function (PRF) is important because it encodes the response of the oscillation to inputs. Recently, the PRF for an inhibitory population of modified theta neurons that generate an ING rhythm was computed by the adjoint method applied to the associated Fokker-Planck equation (FPE) for the model. The modified theta model incorporates conductance-based synapses as well as the voltage and current dynamics. Here, we extended this previous work by creating an excitatory-inhibitory (E-I) network using the modified theta model and described the population dynamics with the corresponding FPE. We conducted a bifurcation analysis of the FPE to find parameter regions which generate gamma oscillations. In order to label the oscillatory parameter regions by their generation mechanisms, we defined ING- and PING-type gamma oscillation in a mathematically plausible way based on the driver of the inhibitory population. We labeled the oscillatory parameter regions by these generation mechanisms and derived PRFs via the adjoint method on the FPE in order to investigate the differences in the responses of each type of oscillation to inputs. PRFs for PING and ING mechanisms are derived and compared. We found the amplitude of the PRF for the excitatory population is larger in the PING case than in the ING case. Finally, the E-I population of the modified theta neuron enabled us to analyze the PRFs of PING-type gamma oscillation and the entrainment ability of E and I populations. We found a parameter region in which PRFs of E and I are both purely positive in the case of PING oscillations. The different entrainment abilities of E and I stimulation as governed by the respective PRFs was compared to direct simulations of finite populations of model neurons. We find that it is easier to entrain the gamma rhythm by stimulating the inhibitory population than by stimulating the excitatory population as has been found experimentally.

Discrete Dynamical Modeling of Influenza Virus Infection Suggests Age-Dependent Differences in Immunity.

Immunosenescence, an age-related decline in immune function, is a major contributor to morbidity and mortality in the elderly. Older hosts exhibit a delayed onset of immunity and prolonged inflammation after an infection, leading to excess damage and a greater likelihood of death. Our study applies a rule-based model to infer which components of the immune response are most changed in an aged host. Two groups of BALB/c mice (aged 12 to 16 weeks and 72 to 76 weeks) were infected with 2 inocula: a survivable dose of 50 PFU and a lethal dose of 500 PFU. Data were measured at 10 points over 19 days in the sublethal case and at 6 points over 7 days in the lethal case, after which all mice had died. Data varied primarily in the onset of immunity, particularly the inflammatory response, which led to a 2-day delay in the clearance of the virus from older hosts in the sublethal cohort. We developed a Boolean model to describe the interactions between the virus and 21 immune components, including cells, chemokines, and cytokines, of innate and adaptive immunity. The model identifies distinct sets of rules for each age group by using Boolean operators to describe the complex series of interactions that activate and deactivate immune components. Our model accurately simulates the immune responses of mice of both ages and with both inocula included in the data (95% accurate for younger mice and 94% accurate for older mice) and shows distinct rule choices for the innate immunity arm of the model between younger and aging mice in response to influenza A virus infection.IMPORTANCE Influenza virus infection causes high morbidity and mortality rates every year, especially in the elderly. The elderly tend to have a delayed onset of many immune responses as well as prolonged inflammatory responses, leading to an overall weakened response to infection. Many of the details of immune mechanisms that change with age are currently not well understood. We present a rule-based model of the intrahost immune response to influenza virus infection. The model is fit to experimental data for young and old mice infected with influenza virus. We generated distinct sets of rules for each age group to capture the temporal differences seen in the immune responses of these mice. These rules describe a network of interactions leading to either clearance of the virus or death of the host, depending on the initial dosage of the virus. Our models clearly demonstrate differences in these two age groups, particularly in the innate immune responses.

Emergent Dynamical Properties of the BCM Learning Rule.

The Bienenstock-Cooper-Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeostatic part of the learning rule depends on the time average of the post-synaptic activity and provides a sliding threshold that distinguishes between increasing or decreasing weights. There are, thus, two essential time scales in the BCM rule: a homeostatic time scale, and a synaptic modification time scale. When the dynamics of the stimulus is rapid enough, it is possible to reduce the BCM rule to a simple averaged set of differential equations. In previous analyses of this model, the time scale of the sliding threshold is usually faster than that of the synaptic modification. In this paper, we study the dynamical properties of these averaged equations when the homeostatic time scale is close to the synaptic modification time scale. We show that instabilities arise leading to oscillations and in some cases chaos and other complex dynamics. We consider three cases: one neuron with two weights and two stimuli, one neuron with two weights and three stimuli, and finally a weakly interacting network of neurons.

Macroscopic phase-resetting curves for spiking neural networks.

The study of brain rhythms is an open-ended, and challenging, subject of interest in neuroscience. One of the best tools for the understanding of oscillations at the single neuron level is the phase-resetting curve (PRC). Synchronization in networks of neurons, effects of noise on the rhythms, effects of transient stimuli on the ongoing rhythmic activity, and many other features can be understood by the PRC. However, most macroscopic brain rhythms are generated by large populations of neurons, and so far it has been unclear how the PRC formulation can be extended to these more common rhythms. In this paper, we describe a framework to determine a macroscopic PRC (mPRC) for a network of spiking excitatory and inhibitory neurons that generate a macroscopic rhythm. We take advantage of a thermodynamic approach combined with a reduction method to simplify the network description to a small number of ordinary differential equations. From this simplified but exact reduction, we can compute the mPRC via the standard adjoint method. Our theoretical findings are illustrated with and supported by numerical simulations of the full spiking network. Notably our mPRC framework allows us to predict the difference between effects of transient inputs to the excitatory versus the inhibitory neurons in the network.

Neurons as oscillators.

Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.