The dynamical behaviour of a population-based rate model with firing adaptation is studied. An excitatory and inhibitory population of neurons is recurrently coupled and a negative feedback term is added to the excitatory population as firing adaptation. In several studies of these models, the UP-DOWN transitions are in focus, which are also exhibited by the investigated model. In this paper we provide a full characterization of equilibrium points with the precise conditions for their existence and stability. Calculations can be performed analytically and explicit formulas can be provided due to the fact that the activation function is a threshold linear function. We use bifurcation analysis to detect significant changes in the phase space. Complete list of bifurcation diagrams is provided with respect to the number of steady states. Oscillatory dynamics of neurobiological relevance are examined through local bifurcation analysis. The study demonstrates that sharp wave-ripple oscillations may emerge in certain regions of the parameter space.
Understanding complex neuronal behavior in our brain requires accurate estimation of neuronal models from observed time-series data. In this study, we propose a data-driven sparse modeling method to estimate multi-dimensional latent variables and electrical properties while extracting essential membrane currents from partially observable time series data. First, we derive a nonlinear state space model from a conductance-based neuron model that couples membrane potential and calcium concentration including a set of candidate membrane currents. Then, we derive a sparse modeling-based expectation-maximization (EM) algorithm. In the expectation step, the expectation of the log-likelihood with Laplace prior distribution is evaluated using the posterior distribution of latent states, which is approximated by a sequential Monte Carlo (SMC) method under one-dimensional noisy observations. In the maximization step, the conductances are estimated as zero value for unnecessary currents whereas those are estimated as non-zero value for necessary currents. We show using simulation with conductance-based neuron model that the proposed method can extract the latent nonlinear neuronal dynamics from partially observable data by estimating latent variables and biophysical parameters and extracting only necessary membrane currents.
Altered connectivity in the prefrontal cortex-amygdala circuit in Major Depressive Disorder is associated with abnormal circuit states. Although substantial evidence from neuroimaging studies has demonstrated that changes in functional connectivity within this circuit during depressive states are commonly reported as a key feature, little is known about how these changes may relate to circuit functioning. In this study, we propose a biophysical computational model that incorporates an improved Jansen-Rit neural mass model to construct the circuit. Resting-state functional magnetic resonance imaging is used to calculate the functional connectivity of the circuit and simulate brain signals. The aim is to investigate how abnormalities in functional connectivity may contribute to or be associated with the underlying mechanisms of circuit dysfunction. The numerical simulation results indicate that under normal conditions, the amygdala plays a key regulatory role in this circuit. When the amygdala is strongly activated in the model, it suppresses the ventromedial prefrontal cortex and activates the rostral anterior cingulate cortex, leading to an abnormal circuit state. In a healthy state, the amygdala maintains activation through bottom-up regulation, supporting emotional regulation and network stability. The amygdala plays a key regulatory role in this circuit, and changes in functional connectivity are a potential factor that may contribute to the imbalance between cognitive control and emotional regulation. We also show within the modeling framework that alterations in functional connectivity are associated with an abnormal circuit state, which may be an important mechanism underlying the expression of depressive symptoms. This study combines data and theoretical modeling techniques, providing a computational perspective on the potential pathological mechanisms of depression and suggests potential therapeutic targets.
Chronic stress is associated with persistent alterations in neural circuit function, yet how these changes are expressed across multiple descriptive levels remains unclear. Here, we re-analyzed in vivo GCaMP8s recordings from BLA-DMS and CeA-DMS projection pathways using complementary distributional, dynamical, and computational approaches. Distributional analyses based on Kullback-Leibler (KL) divergence revealed stress-associated changes in the temporal organization of neural activity patterns, particularly following acute aversive perturbations where stressed animals exhibited prolonged deviations from baseline distributions. Dynamical analyses using phenomenological second-order regression models revealed corresponding alterations in recovery-related properties. Following footshock, stress was associated with shifts in principal eigenvalue distributions and reconstructed quasi-potential profiles, whereas learned lever press–reward behaviors exhibited broadly similar local stability despite differences in coefficient structure. Notably, stress-related differences in reconstructed dynamics were detectable during task acquisition even when overt behavioral performance remained comparable between groups. To examine whether these qualitative signatures could arise from simple computational principles, we implemented a minimal artificial neural network (ANN) framework as a hypothesis-generating sufficiency test. Models incorporating asymmetric optimization objectives reproduced selected experimental signatures, whereas symmetric objectives did not. Together, these findings suggest that chronic stress is associated with a reorganization of neural activity patterns across multiple descriptive levels, altering responses to perturbations and producing detectable changes in neural activity organization before overt behavioral divergence emerges.
Cognitive neural circuits must balance the plasticity needed for continual learning with the stability needed to preserve an underlying reasoning framework, called a schema. How circuit learning rules form and protect such schemata from being continually overwritten during learning remains unknown. On a visual boundary detection task, we show that a hierarchical learning algorithm creates an invariant schema circuit whose weights remain fixed following sparse initial training, with additional data serving to refine the upstream representation. In contrast, the end-to-end backpropagation algorithm used to train nearly all current artificial neural networks comprehensively changes its weights throughout training. We show that the hierarchical schema makes an independent mechanistic hypothesis about circuit computation that is consistent with experimental ordering observed in single-neuron measurements in humans. These results suggest that hierarchical learning is sufficient to encode biologically consistent persistent cognitive models within otherwise malleable neural networks.
Cortical circuits exhibit variable yet bounded activity patterns, suggesting operation near—but not within—fully chaotic regimes. Here we develop a minimal three-variable rate model for primary visual cortex (V1) that reveals how biologically motivated feedback mechanisms can function as intrinsic chaos controllers. We adopt the simplest known chaotic Lotka–Volterra system as a phenomenological scaffold and introduce three biologically motivated modifications: excitatory-to-inhibitory (E → I) feedback coupling, homeostatic regulation of modulatory drive, and orientation-tuned sensory input. These modifications transform the excitatory (E), inhibitory (I), and modulatory (M) population dynamics from chaotic strange attractors into controlled limit cycles—a 93 = 0.38 ± 0.09 ), stimulus-induced variability quenching, and realistic spiking irregularity when coupled to Hodgkin–Huxley neurons (CV _ISI = 0.27 , within in vitro range). Parameter space analysis reveals that feedback mechanisms robustly stabilize activity across most of the tested chaotic regime. We further demonstrate that, within this minimal structure, the specific (1-I^2) disinhibition nonlinearity enables chaos—bounded alternatives tested do not support chaotic dynamics. Our findings suggest that cortical circuits possess an intrinsic capacity for chaos that is actively suppressed by canonical feedback motifs, positioning the brain at the edge of instability where computational flexibility meets reliable signal processing.
Gluatamatergic synaptic transmission, critical for learning and memory, relies on precise regulation of extracellular glutamate levels by astrocytic transporters, particularly EAAT2. While existing models of AMPA and NMDA receptor kinetics often oversimplify glutamate dynamics or become computationally intractable, this study develops a balanced, biophysically grounded model that integrates glutamate transport, receptor sensitivity, and electrotonic effects. Using rat hippocampal slices, we recorded postsynaptic currents in CA1 pyramidal neurons under control conditions and during glutamate transporter blockade. The proposed mathematical model, formulated as a system of seven ordinary differential equations, distinguishes somatic and dendritic compartments, synaptic plasticity, and differential glutamate sensitivity of AMPA and NMDA receptors. Key findings reveal that the glutamate transporter blockade prolongs NMDA receptor-mediated currents without altering AMPA receptor kinetics, consistent with the higher glutamate sensitivity of NMDA receptors. The model also predicts glutamate concentrations in synaptic and extrasynaptic spaces, offering insights into spatial neurotransmitter dynamics. Furthermore, it accounts for voltage-dependent NMDA responses and short-term plasticity observed experimentally. By bridging the gap between oversimplified and overly complex approaches, this work provides a versatile tool for studying synaptic transmission in normal and pathological conditions, such as epilepsy or neurodegenerative diseases, where glutamate dysregulation plays a central role.
This article presents a computational study of the Hodgkin-Huxley model and the simulation of action potential propagation in an unmyelinated axon using the finite volume method. By implementing a voltage-clamp patch and tracking detailed ionic and capacitive currents and channel gating, we achieve robust and unit-consistent simulation of action potential initiation and propagation.
The brain operates across multiple spatial and temporal scales, necessitating computationally efficient models that link micro-scale mechanisms to meso- and macro-scale dynamics. Here, we introduce a novel convolutional neural mass model (CNMM) that computes the meso-scale activity of spatially discretized neural populations (“neural masses") in the rat hippocampal CA3 subregion. The CNMM employs a kernel-based architecture, leveraging first-order Volterra expansions with Laguerre (temporal) and Chebyshev (spatial) basis functions to transform input spike densities from entorhinal cortex (EC), dentate gyrus (DG), and neighboring CA3 masses into output CA3 spike density. The model was trained and validated using data from a biophysically detailed large-scale mechanistic model (LSM) simulating exploratory behavior. The CNMM achieved high predictive accuracy for spike density across 32 neural masses spanning the entire extent of CA3 (mean correlation coefficient R = 0.951 ± 0.027 ) and replicated theta and beta oscillations consistent with experimental findings. When extended for forward modeling, the CNMM accurately predicted local field potentials (LFPs) at a single neural mass ( R = 0.952 ), demonstrating feasibility of this approach. Kernel analysis revealed topographic gradients in afferent integration, with DG inputs dominating proximally (CA3c) and associational connections distally (CA3a), aligning with anatomical gradients. Compared to the LSM, the CNMM provided a 658-fold speedup in simulation time, 322-fold reduction in memory usage, and 183-fold less disk space for LFP predictions. This framework offers a scalable, efficient approach for meso-scale modeling of neural tissue, bridging detailed simulations with empirical data and laying groundwork for future investigations into both normal and pathological brain function.
In this paper we present calculations of the entropy and information transmission associated with spike trains produced by the circle/circle bursting model neuron in response to filtered white-noise stimuli. For most computations, we treated the bursts as unitary objects and estimated the entropy from the time intervals between the first spikes in consecutive bursts. In one case, we considered the intervals associated with all the spikes in the burst train or only the first and last spikes of each burst. We found that the entropy per burst was maximized when the stimulus was well matched to the neuron’s natural burst frequency and that the entropy/spike increased considerably when the duration of the burst was considered. Moreover, for a noisy stimulus for which the deterministic part of the stimulus was close to the natural burst frequency but most of the noise was at much higher frequencies, the bursting neuron transmitted considerably more information in bits/spike than a spiking model neuron constructed using similar ionic conductances.
Image-computable models of primate retinal ganglion cell (RGC) mosaics that are synthesized and constrained jointly by optical, anatomical and physiological properties, and which operate on images defined by their spatial-spectral radiance, do not currently exist. Here, we deploy a novel computational framework which synthesizes mosaics of linear spatio-chromatic receptive fields (RFs) of ON midget RGCs (mRGCs) by integrating published anatomical, physiological, and optical quality measurements, all varying with eccentricity. We use the synthesized mRGC mosaics to simulate both in vivo and in vitro physiological experiments and demonstrate the model’s consistency with published data. The model enables computation of how visual performance is shaped by the representation of visual information provided by the linear spatiochromatic processing stage of midget RGCs. The developed computational framework carefully accounts for the effect of physiological optics on mRGC responses, enables comparison of in vivo and in vitro data, and allows exploration of how different assumptions about RF organization, such as selectivity for the type of cones pooled by the RF center mechanism, affect physiological responses and psychophysical performance. The open-source and freely available implementation provides a platform for understanding how the linear spatiochromatic receptive field representation of the mRGCs shapes visual performance, as well as a foundation for future work that incorporates response nonlinearities, temporal filtering, and extends to additional RGC mosaics.
A key component to understanding the brain is determining the influence of groups of neurons on each other relative to other influences. In the brain, all neurons are driven by the activity of other neurons, some of which may be simultaneously recorded, but most are not. As such, models of neuronal activity need to account for simultaneously recorded neurons and the influences of unmeasured neurons. This can be done through the inclusion of model terms for observed external variables (e.g., tuning to stimuli), observed internal variables (e.g., coupling to recorded neural activities), as well as terms for latent sources of variability. Despite broad utilization, however, evaluation of systematic errors during inference is rarely performed, and sources of systematic error are poorly understood. Through extensive numerical study and analytic calculation, we show that common inference procedures for static and dynamic models typically have systematic errors. Counter to common intuition, we found that model non-identifiability contributes to systematic errors in parameter estimation, not variance inflation, making it a particularly insidious form of statistical error. We demonstrate that accurate parameter selection before estimation resolves model non-identifiability and mitigates the associated systematic errors. In diverse neurophysiology data sets (multiple single unit recordings in primary visual cortex and hippocampus, ECoG from primary auditory cortex), we found that common methods typically overestimate the contributions of interactions between neurons, while the influence of exogenous variables is underestimated. Essentially, when there are positive correlations due to unobserved shared variability, estimated coupling amongst neurons will be inflated at the expense of tuning. We explain heterogeneity in observed systematic errors across neurophysiology data sets in terms of data statistics and experimental design. Together, our results identify the causes of statistical errors in structural equation models of simultaneous systems with endogenous, exogenous, and latent variables, provide inference procedures to mitigate those errors, and reveal and explain the impact of those errors in diverse neural data sets.
