Conductance-based refractory density model of primary visual cortex

Response retention and apparent motion effect in visual cortex models

Apparent motion is a visual illusion in which stationary stimuli, flashing in distinct spatial locations at certain time intervals, are perceived as one stimulus moving between these locations. In the primary visual cortex, apparent-motion stimuli produce smooth spatio-temporal patterns of activity similar to those produced by continuously moving stimuli. An important prerequisite for producing such activity patterns is prolongation of responses to brief stimuli. Indeed, a brief stimulus can evoke in the visual cortex a long response, outlasting the stimulus by hundreds of milliseconds. Here we use firing-rate based models with simple ring structure, and biologically-detailed conductance-based refractory density (CBRD) model with retinotopic space representation to analyze the response retention and the origin of smooth profiles of activity in response to apparent-motion stimuli. We show that the strength of recurrent connectivity is the major factor that endorses neuronal networks with the ability for response retention. The same strengths of recurrent connections mediate the appearance of bump attractor in the ring models. Factors such as synaptic depression, NMDA receptor mediated currents, and conductances regulating spike adaptation influence response retention, but cannot substitute for the weakness of recurrent connections to reproduce response retention in models with weak connectivity. However, the weakness of lateral recurrent connections can be compensated by layering: in multi-layer models even with weaker connections the activity retains due to its feedforward propagation from layer to layer. Using CBRD model with retinotopic space representation we further show that smooth spatio-temporal profiles of activity in response to apparent-motion stimuli are produced in the models expressing response retention, but not in the models that fail to produce response retention. Together, these results demonstrate a link between response retention and the ability of neuronal networks to generate spatio-temporal patterns of activity, which are compatible with perception of apparent motion.

Information Encoding in Bursting Spiking Neural Network Modulated by Astrocytes

We investigated a mathematical model composed of a spiking neural network (SNN) interacting with astrocytes. We analysed how information content in the form of two-dimensional images can be represented by an SNN in the form of a spatiotemporal spiking pattern. The SNN includes excitatory and inhibitory neurons in some proportion, sustaining the excitation-inhibition balance of autonomous firing. The astrocytes accompanying each excitatory synapse provide a slow modulation of synaptic transmission strength. An information image was uploaded to the network in the form of excitatory stimulation pulses distributed in time reproducing the shape of the image. We found that astrocytic modulation prevented stimulation-induced SNN hyperexcitation and non-periodic bursting activity. Such homeostatic astrocytic regulation of neuronal activity makes it possible to restore the image supplied during stimulation and lost in the raster diagram of neuronal activity due to non-periodic neuronal firing. At a biological point, our model shows that astrocytes can act as an additional adaptive mechanism for regulating neural activity, which is crucial for sensory cortical representations.

A framework for macroscopic phase-resetting curves for generalised spiking neural networks

Brain rhythms emerge from synchronization among interconnected spiking neurons. Key properties of such rhythms can be gleaned from the phase-resetting curve (PRC). Inferring the PRC and developing a systematic phase reduction theory for large-scale brain rhythms remains an outstanding challenge. Here we present a theoretical framework and methodology to compute the PRC of generic spiking networks with emergent collective oscillations. We adopt a renewal approach where neurons are described by the time since their last action potential, a description that can reproduce the dynamical feature of many cell types. For a sufficiently large number of neurons, the network dynamics are well captured by a continuity equation known as the refractory density equation. We develop an adjoint method for this equation giving a semi-analytical expression of the infinitesimal PRC. We confirm the validity of our framework for specific examples of neural networks. Our theoretical framework can link key biological properties at the individual neuron scale and the macroscopic oscillatory network properties. Beyond spiking networks, the approach is applicable to a broad class of systems that can be described by renewal processes.

The formation of oscillatory neuronal assemblies at the network level has been hypothesized to be fundamental to many cognitive and motor functions. One prominent tool to understand the dynamics of oscillatory activity response to stimuli, and hence the neural code for which it is a substrate, is a nonlinear measure called Phase-Resetting Curve (PRC). At the network scale, the PRC defines the measure of how a given synaptic input perturbs the timing of next upcoming volley of spike assemblies: either advancing or delaying this timing. As a further application, one can use PRCs to make unambiguous predictions about whether communicating networks of neurons will phase-lock as it is often observed across the cortical areas and what would be this stable phase-configuration: synchronous, asynchronous or with asymmetric phase-shifts. The latter configuration also implies a preferential flow of information form the leading network to the follower, thereby giving causal signatures of directed functional connectivity. Because of the key position of the PRC in studying synchrony, information flow and entrainment to external forcing, it is crucial to move toward a theory that allows to compute the PRCs of network-wide oscillations not only for a restricted class of models, as has been done in the past, but to network descriptions that are generalized and can reflect flexibly single cell properties. In this manuscript, we tackle this issue by showing how the PRC for network oscillations can be computed using the adjoint systems of partial differential equations that define the dynamics of the neural activity density.

Ictal wavefront propagation in slices and simulations with conductance-based refractory density model

The mechanisms determining ictal discharge (ID) propagation are still not clear. In the present study, we aimed to examine these mechanisms in animal and mathematical models of epileptiform activity. Using double-patch and extracellular potassium ion concentration recordings in rat hippocampal-cortical slices, we observed that IDs moved at a speed of about 1 mm/s or less. The mechanisms of such slow propagation have been studied with a mathematical, conductance-based refractory density (CBRD) model that describes the GABA- and glutamatergic neuronal populations’ interactions and ion dynamics in brain tissue. The modeling study reveals two main factors triggerring IDs: (i) increased interneuronal activity leading to chloride ion accumulation and a consequent depolarizing GABAergic effect and (ii) the elevation of extracellular potassium ion concentration. The local synaptic transmission followed by local potassium ion extrusion and GABA receptor-mediated chloride ion accumulation underlies the ID wavefront’s propagation. In contrast, potassium ion diffusion in the extracellular space is slower and does not affect ID’s speed. The short discharges, constituting the ID, propagate much faster than the ID front. The accumulation of sodium ions inside neurons due to their hyperactivity and glutamatergic currents boosts the Na⁺/K⁺ pump, which terminates the ID. Knowledge of the mechanism of ID generation and propagation contributes to the development of new treatments against epilepsy.

During an epileptic seizure, neuronal excitation spreads across the brain tissue and is accompanied by significant changes in ionic concentrations. Ictal discharge front spreads at low speeds, less than 1 mm/s. Mechanisms underlying this phenomenon are not yet well understood. We study these mechanisms using electrophysiological recordings in brain slices and computer simulations. Our detailed biophysical model describing neuronal populations’ interaction, spatial propagation, and ionic dynamics reproduces the generation and propagation of spontaneously repeating ictal discharges. The simulations are consistent with our recordings of the electrical activity and the extracellular potassium ion concentration. We distinguished between the two alternative mechanisms of the ictal wavefront propagation: (i) the diffusion of potassium ions released from excited neurons, which depolarizes distant neurons and thus supports excitation, and (ii) the axonal spread of excitation followed by the local extracellular potassium ion accumulation that supports the excitation. Our simulations provide evidence in favor of the latter mechanism. Our experiment-based modeling contributes to a mathematical description of brain tissue functioning and potentially contributes to developing new treatments against epilepsy.

A strategy for mapping biophysical to abstract neuronal network models applied to primary visual cortex

A fundamental challenge for the theoretical study of neuronal networks is to make the link between complex biophysical models based directly on experimental data, to progressively simpler mathematical models that allow the derivation of general operating principles. We present a strategy that successively maps a relatively detailed biophysical population model, comprising conductance-based Hodgkin-Huxley type neuron models with connectivity rules derived from anatomical data, to various representations with fewer parameters, finishing with a firing rate network model that permits analysis. We apply this methodology to primary visual cortex of higher mammals, focusing on the functional property of stimulus orientation selectivity of receptive fields of individual neurons. The mapping produces compact expressions for the parameters of the abstract model that clearly identify the impact of specific electrophysiological and anatomical parameters on the analytical results, in particular as manifested by specific functional signatures of visual cortex, including input-output sharpening, conductance invariance, virtual rotation and the tilt after effect. Importantly, qualitative differences between model behaviours point out consequences of various simplifications. The strategy may be applied to other neuronal systems with appropriate modifications.

A hierarchy of theoretical approaches to study a neuronal network depends on a tradeoff between biological fidelity and mathematical tractibility. Biophysically-detailed models consider cellular mechanisms and anatomically defined synaptic circuits, but are often too complex to reveal insights into fundamental principles. In contrast, increasingly abstract reduced models facilitate analytical insights. To better ground the latter to the underlying biology, we describe a systematic procedure to move across the model hierarchy that allows understanding how changes in biological parameters—physiological, pathophysiological, or because of new data—impact the behaviour of the network. We apply this approach to mammalian primary visual cortex, and examine how the different models in the hierarchy reproduce functional signatures of this area, in particular the tuning of neurons to the orientation of a visual stimulus. Our work provides a navigation of the complex parameter space of neural network models faithful to biology, as well as highlighting how simplifications made for mathematical convenience can fundamentally change their behaviour.

Refractory density model of cortical direction selectivity: Lagged-nonlagged, transient-sustained, and On-Off thalamic neuron-based mechanisms and intracortical amplification

A biophysically detailed description of the mechanisms of the primary vision is still being developed. We have incorporated a simplified, filter-based description of retino-thalamic visual signal processing into the detailed, conductance-based refractory density description of the neuronal population activity of the primary visual cortex. We compared four mechanisms of the direction selectivity (DS), three of them being based on asymmetrical projections of different types of thalamic neurons to the cortex, distinguishing between (i) lagged and nonlagged, (ii) transient and sustained, and (iii) On and Off neurons. The fourth mechanism implies a lack of subcortical bias and is an epiphenomenon of intracortical interactions between orientation columns. The simulations of the cortical response to moving gratings have verified that first three mechanisms provide DS to an extent compared with experimental data and that the biophysical model realistically reproduces characteristics of the visual cortex activity, such as membrane potential, firing rate, and synaptic conductances. The proposed model reveals the difference between the mechanisms of both the intact and the silenced cortex, favoring the second mechanism. In the fourth case, DS is weaker but significant; it completely vanishes in the silenced cortex.DS in the On-Off mechanism derives from the nonlinear interactions within the orientation map. Results of simulations can help to identify a prevailing mechanism of DS in V1. This is a step towards a comprehensive biophysical modeling of the primary visual system in the frameworks of the population rate coding concept.

A major mechanism that underlies tuning of cortical neurons to the direction of a moving stimulus is still debated. Considering the visual cortex structured with orientation-selective columns, we have realized and compared in our biophysically detailed mathematical model four hypothetical mechanisms of the direction selectivity (DS) known from experiments. The present model accomplishes our previous model that was tuned to experimental data on excitability in slices and reproduces orientation tuning effects in vivo. In simulations, we have found that the convergence of inputs from so-called transient and sustained (or lagged and nonlagged) thalamic neurons in the cortex provides an initial bias for DS, whereas cortical interactions amplify the tuning. In the absence of any bias, DS emerges as an epiphenomenon of the orientation map. In the case of a biased convergence of On- and Off- thalamic inputs, DS emerges with the help of the intracortical interactions on the orientation map, also. Thus, we have proposed a comprehensive description of the primary vision and revealed characteristic features of different mechanisms of DS in the visual cortex with columnar structure.

Mind the last spike - firing rate models for mesoscopic populations of spiking neurons

The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level. We review recent advances achieved by the RDM to obtain efficient population density equations for networks of generalized integrate-and-fire (GIF) neurons - a class of neuron models that has been successfully fitted to various cell types. The theory not only predicts the nonstationary dynamics of large populations of neurons but also permits an extension to finite-size populations and a systematic reduction to low-dimensional rate dynamics. The new types of rate models will allow a re-examination of models of cortical computations under biological constraints.

Conductance-Based Refractory Density Approach for a Population of Bursting Neurons

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modelling interacting populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study, we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting population model makes use of slow-fast analysis, which leads to a novel methodology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explorations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).

Mathematical model of Na-K-Cl homeostasis in ictal and interictal discharges

Despite big experimental data on the phenomena and mechanisms of the generation of ictal and interictal discharges (IDs and IIDs), mathematical models that can describe the synaptic interactions of neurons and the ionic dynamics in biophysical detail are not well-established. Based on experimental recordings of combined hippocampal-entorhinal cortex slices from rats in a high-potassium and a low-magnesium solution containing 4-aminopyridine as well as previous observations of similar experimental models, this type of mathematical model has been developed. The model describes neuronal excitation through the application of the conductance-based refractory density approach for three neuronal populations: two populations of glutamatergic neurons with hyperpolarizing and depolarizing GABAergic synapses and one GABAergic population. The ionic dynamics account for the contributions of voltage-gated and synaptic channels, active and passive transporters, and diffusion. The relatively slow dynamics of potassium, chloride, and sodium ion concentrations determine the transitions from pure GABAergic IIDs to IDs and GABA-glutamatergic IIDs. The model reproduces different types of IIDs, including those initiated by interneurons; repetitive IDs; tonic and bursting modes of an ID composed of clustered IID-like events. The simulations revealed contributions from different ionic channels to the ion concentration dynamics before and during ID generation. The proposed model is a step forward to an optimal mathematical description of the mechanisms of epileptic discharges.

Spatial propagation of interictal discharges along the cortex

Interictal discharges (IIDs) accompany epileptic seizures and highlight the mechanisms of pathological activity. The propagation of IIDs along the neural tissue is not well understood. To simulate IID propagation, this study proposes a new mathematical model that uses the conductance-based refractory density approach for glutamatergic and GABAergic neuronal populations. The mathematical model is found to be consistent with experimental double-patch registrations in the 4-aminopyridine in vitro model of epilepsy. In slices, the spontaneous activity of interneurons leads to their synchronization by means of the depolarizing GABAmediated response, thus initiating IIDs. Modeling reveals a clustering of interneuronal synchronization followed by IIDs with activity fronts that propagate along the cortex. The GABA-mediated depolarization either remains to be subthreshold for the principal neurons and thus results in pure GABAergic IIDs (IID1s) or leads to glutamatergic excitation, thus resulting in another type of IIDs (IID2s). In both the model and experiment, IIDs propagate as waves, with constant activity profiles and velocity. The speed of IIDs is of the order of tens of mm/s and is larger for IID2s than for IID1s (40 and 20 mm/s, respectively). The simulations, consistent with experimental observations, show that the wavelike propagation of IIDs initiated by interneurons is determined by local synaptic connectivity under the conditions of depolarizing GABA.

Seizure-Induced Potentiation of AMPA Receptor-Mediated Synaptic Transmission in the Entorhinal Cortex

Excessive excitation is considered one of the key mechanisms underlying epileptic seizures. We investigated changes in the evoked postsynaptic responses of medial entorhinal cortex (ERC) pyramidal neurons by seizure-like events (SLEs), using the modified 4-aminopyridine (4-AP) model of epileptiform activity. Rat brain slices were perfused with pro-epileptic solution contained 4-AP and elevated potassium and reduced magnesium concentration. We demonstrated that 15-min robust epileptiform activity in slices leads to an increase in the amplitude of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR)-mediated component of the evoked response, as well as an increase in the polysynaptic γ-aminobutyric acid (GABA) and N-methyl-D-aspartate (NMDA) receptor-mediated components. The increase in AMPA-mediated postsynaptic conductance depends on NMDA receptor activation. It persists for at least 15 min after the cessation of SLEs and is partly attributed to the inclusion of calcium-permeable AMPA receptors in the postsynaptic membrane. The mathematical modeling of the evoked responses using the conductance-based refractory density (CBRD) approach indicated that such augmentation of the AMPA receptor function and depolarization by GABA receptors results in prolonged firing that explains the increase in polysynaptic components, which contribute to overall network excitability. Taken together, our data suggest that AMPA receptor enhancement could be a critical determinant of sustained status epilepticus (SE).

Adaptation and Inhibition Control Pathological Synchronization in a Model of Focal Epileptic Seizure

Pharmacoresistant epilepsy is a common neurological disorder in which increased neuronal intrinsic excitability and synaptic excitation lead to pathologically synchronous behavior in the brain. In the majority of experimental and theoretical epilepsy models, epilepsy is associated with reduced inhibition in the pathological neural circuits, yet effects of intrinsic excitability are usually not explicitly analyzed. Here we present a novel neural mass model that includes intrinsic excitability in the form of spike-frequency adaptation in the excitatory population. We validated our model using local field potential (LFP) data recorded from human hippocampal/subicular slices. We found that synaptic conductances and slow adaptation in the excitatory population both play essential roles for generating seizures and pre-ictal oscillations. Using bifurcation analysis, we found that transitions towards seizure and back to the resting state take place via Andronov-Hopf bifurcations. These simulations therefore suggest that single neuron adaptation as well as synaptic inhibition are responsible for orchestrating seizure dynamics and transition towards the epileptic state.

Conductance-based refractory density approach: comparison with experimental data and generalization to lognormal distribution of input current

The conductance-based refractory density (CBRD) approach is an efficient tool for modeling interacting neuronal populations. The model describes the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons, each receiving individual Gaussian noise and a common time-varying deterministic input. However, the approach requires experimental validation and extension to cases of distributed input signals (or input weights) among different neurons of such an ensemble. Here the CBRD model is verified by comparing with experimental data and then generalized for a lognormal (LN) distribution of the input weights. The model with equal weights is shown to reproduce efficiently the post-spike time histograms and the membrane voltage of experimental multiple trial response of single neurons to a step-wise current injection. The responses reveal a more rapid reaction of the firing-rate than voltage. Slow adaptive potassium channels strongly affected the shape of the responses. Next, a computationally efficient CBRD model is derived for a population with the LN input weight distribution and is compared with the original model with equal input weights. The analysis shows that the LN distribution: (1) provides a faster response, (2) eliminates oscillations, (3) leads to higher sensitivity to weak stimuli, and (4) increases the coefficient of variation of interspike intervals. In addition, a simplified firing-rate type model is tested, showing improved precision in the case of a LN distribution of weights. The CBRD approach is recommended for complex, biophysically detailed simulations of interacting neuronal populations, while the modified firing-rate type model is recommended for computationally reduced simulations.

Computational model of interictal discharges triggered by interneurons

Interictal discharges (IIDs) are abnormal waveforms registered in the periods before or between seizures. IIDs that are initiated by GABAergic interneurons have not been mathematically modeled yet. In the present study, a mathematical model that describes the mechanisms of these discharges is proposed. The model is based on the experimental recordings of IIDs in pyramidal neurons of the rat entorhinal cortex and estimations of synaptic conductances during IIDs. IIDs were induced in cortico-hippocampal slices by applying an extracellular solution with 4-aminopyridine, high potassium, and low magnesium concentrations. Two different types of IIDs initiated by interneurons were observed. The first type of IID (IID1) was pure GABAergic. The second type of IID (IID2) was induced by GABAergic excitation and maintained by recurrent interactions of both GABA- and glutamatergic neuronal populations. The model employed the conductance-based refractory density (CBRD) approach, which accurately approximates the firing rate of a population of similar Hodgkin-Huxley-like neurons. The model of coupled excitatory and inhibitory populations includes AMPA, NMDA, and GABA-receptor-mediated synapses and gap junctions. These neurons receive both arbitrary deterministic input and individual colored Gaussian noise. Both types of IIDs were successfully reproduced in the model by setting two different depolarized levels for GABA-mediated current reversal potential. It was revealed that short-term synaptic depression is a crucial factor in ceasing each of the discharges, and it also determines their durations and frequencies.

Improving voltage-sensitive dye imaging: with a little help from computational approaches

Voltage-sensitive dye imaging (VSDI) is a key neurophysiological recording tool because it reaches brain scales that remain inaccessible to other techniques. The development of this technique from in vitro to the behaving nonhuman primate has only been made possible thanks to the long-lasting, visionary work of Amiram Grinvald. This work has opened new scientific perspectives to the great benefit to the neuroscience community. However, this unprecedented technique remains largely under-utilized, and many future possibilities await for VSDI to reveal new functional operations. One reason why this tool has not been used extensively is the inherent complexity of the signal. For instance, the signal reflects mainly the subthreshold neuronal population response and is not linked to spiking activity in a straightforward manner. Second, VSDI gives access to intracortical recurrent dynamics that are intrinsically complex and therefore nontrivial to process. Computational approaches are thus necessary to promote our understanding and optimal use of this powerful technique. Here, we review such approaches, from computational models to dissect the mechanisms and origin of the recorded signal, to advanced signal processing methods to unravel new neuronal interactions at mesoscopic scale. Only a stronger development of interdisciplinary approaches can bridge micro- to macroscales.

Simplest relationship between local field potential and intracellular signals in layered neural tissue

The relationship between the extracellularly measured electric field potential resulting from synaptic activity in an ensemble of neurons and intracellular signals in these neurons is an important but still open question. Based on a model neuron with a cylindrical dendrite and lumped soma, we derive a formula that substantiates a proportionality between the local field potential and the total somatic transmembrane current that emerges from the difference between the somatic and dendritic membrane potentials. The formula is tested by intra- and extracellular recordings of evoked synaptic responses in hippocampal slices. Additionally, the contribution of different membrane currents to the field potential is demonstrated in a two-population mean-field model. Our formalism, which allows for a simple estimation of unknown dendritic currents directly from somatic measurements, provides an interpretation of the local field potential in terms of intracellularly measurable synaptic signals. It is also applicable to the study of cortical activity using two-compartment neuronal population models.

A mathematical model of color and orientation processing in V1

Orientation processing in the primary visual cortex (V1) has been experimentally investigated in detail and reproduced in models, while color processing remains unclear. Thus, we have constructed a mathematical model of color and orientation processing in V1. The model is mainly based on the following experimental evidence concerning color blobs: A blob contains overlapping neuronal patches activated by different hues, so that each blob represents a full gamut of hue and might be structured with a loop (Xiao et al. in NeuroImage 35:771-786, 2007). The proposed model describes a set of orientation hypercolumns and color blobs, in which color and orientation preferences are represented by the poloidal and toroidal angles of a torus, correspondingly. The model consists of color-insensitive (CI) and color-sensitive (CS) neuronal populations, which are described by a firing-rate model. The set of CI neurons is described by the classical ring model (Ben-Yishai et al. in Proc Natl Acad Sci USA 92:3844-3848, 1995) with recurrent connections in the orientation space; similarly, the set of CS neurons is described in the color space and also receives input from CI neurons of the same orientation preference. The model predictions are as follows: (1) responses to oriented color stimuli are significantly stronger than those to non-oriented color stimuli; (2) the activity of CS neurons in total is higher than that of CI neurons; (3) a random color can be illusorily perceived in the case of gray oriented stimulus; (4) in response to two-color stimulus in the marginal phase, the network chooses either one of the colors or the intermediate color; (5) input to a blob has rather continual representation of a hue than discrete one (with two narrowly tuned opponent signals).