The Determining Role of Covariances in Large Networks of Stochastic Neurons.

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity.

Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition.

Fifty years ago, Wilson and Cowan developed a mathematical model to describe the activity of neural populations. In this seminal work, they divided the cells in three groups: active, sensitive and refractory, and obtained a dynamical system to describe the evolution of the average firing rates of the populations. In the present work, we investigate the impact of the often neglected refractory state and show that taking it into account can introduce new dynamics. Starting from a continuous-time Markov chain, we perform a rigorous derivation of a mean-field model that includes the refractory fractions of populations as dynamical variables. Then, we perform bifurcation analysis to explain the occurrence of periodic solutions in cases where the classical Wilson-Cowan does not predict oscillations. We also show that our mean-field model is able to predict chaotic behavior in the dynamics of networks with as little as two populations.

Sensitivity of the electrical response of a node of Ranvier model to alterations of the myelin sheath geometry.

Nodes of Ranvier play critical roles in the generation and transmission of action potentials. Alterations in node properties during pathology and/or development are known to affect the speed and quality of electrical transmission. From a modelling standpoint, nodes of Ranvier are often described by systems of ordinary differential equations neglecting or greatly simplifying their geometric structure. These approaches fail to accurately describe how fine scale alteration in the node geometry or in myelin thickness in the paranode region will impact action potential generation and transmission. Here, we rely on a finite element approximation to describe the three dimensional geometry of a node of Ranvier. With this, we are able to investigate how sensitive is the electrical response to alterations in the myelin sheath and paranode geometry. We could in particular investigate irregular loss of myelin, which might be more physiologically relevant than the uniform loss often described through simpler modelling approaches.

Modelling dendritic spines with the finite element method, investigating the impact of geometry on electric and calcic responses.

Understanding the relationship between shape and function of dendritic spines is an elusive topic. Several modelling approaches have been used to investigate the interplay between spine geometry, calcium diffusion and electric signalling. We here use a second order finite element method to solve the Poisson-Nernst-Planck equations and describe electrodiffusion in dendritic spines. With this, we obtain relationships between dendritic geometry and calcic as well as electric responses to synaptic events. Our findings support the hypothesis that spine geometry plays a role shaping the electrical responses to synaptic events. Our method was also able to reveal the fine scale distribution of calcium in spines with irregular shapes.

Sensitivity analysis of the Poisson Nernst-Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model.

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst-Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

Allosteric modulation of metabotropic glutamate receptors by chloride ions.

Metabotropic glutamate receptors (mGluRs) play key roles in the modulation of many synapses. Chloride (Cl(-)) is known to directly bind and regulate the function of different actors of neuronal activity, and several studies have pointed to the possible modulation of mGluRs by Cl(-). Herein, we demonstrate that Cl(-) behaves as a positive allosteric modulator of mGluRs. For example, whereas glutamate potency was 3.08 ± 0.33 µM on metabotropic glutamate (mGlu) 4 receptors in high-Cl(-) buffer, signaling activity was almost abolished in low Cl(-) in cell-based assays. Cl(-) potency was 78.6 ± 3.5 mM. Cl(-) possesses a high positive cooperativity with glutamate (Hill slope ≈6 on mGlu4), meaning that small variations in [Cl(-)] lead to large variations in glutamate action. Using molecular modeling and mutagenesis, we have identified 2 well-conserved Cl(-) binding pockets in the extracellular domain of mGluRs. Moreover, modeling of activity-dependent Cl(-) variations at GABAergic synapses suggests that these variations may be compatible with a dynamic modulation of the most sensitive mGluRs present in these synapses. Taken together, these data reveal a necessary role of Cl(-) for the glutamate activation of many mGluRs. Exploiting Cl(-) binding pockets may yield to the development of innovative regulators of mGluR activity.

A scoping review of mathematical models covering Alzheimer's disease progression.

Alzheimer's disease is a complex, multi-factorial, and multi-parametric neurodegenerative etiology. Mathematical models can help understand such a complex problem by providing a way to explore and conceptualize principles, merging biological knowledge with experimental data into a model amenable to simulation and external validation, all without the need for extensive clinical trials. We performed a scoping review of mathematical models describing the onset and evolution of Alzheimer's disease as a result of biophysical factors following the PRISMA standard. Our search strategy applied to the PubMed database yielded 846 entries. After using our exclusion criteria, only 17 studies remained from which we extracted data, which focused on three aspects of mathematical modeling: how authors addressed continuous time (since even when the measurements are punctual, the biological processes underlying Alzheimer's disease evolve continuously), how models were solved, and how the high dimensionality and non-linearity of models were managed. Most articles modeled Alzheimer's disease at the cellular level, operating on a short time scale (e.g., minutes or hours), i.e., the micro view (12/17); the rest considered regional or brain-level processes with longer timescales (e.g., years or decades) (the macro view). Most papers were concerned primarily with amyloid beta (n = 8), few described both amyloid beta and tau proteins (n = 3), while some considered more than these two factors (n = 6). Models used partial differential equations (n = 3), ordinary differential equations (n = 7), and both partial differential equations and ordinary differential equations (n = 3). Some did not specify their mathematical formalism (n = 4). Sensitivity analyses were performed in only a small number of papers (4/17). Overall, we found that only two studies could be considered valid in terms of parameters and conclusions, and two more were partially valid. This puts the majority (n = 13) as being either invalid or with insufficient information to ascertain their status. This was the main finding of our paper, in that serious shortcomings make their results invalid or non-reproducible. These shortcomings come from insufficient methodological description, poor calibration, or the impossibility of experimentally validating or calibrating the model. Those shortcomings should be addressed by future authors to unlock the usefulness of mathematical models in Alzheimer's disease.

A computational model of Alzheimer's disease at the nano, micro, and macroscales.

Introduction: Mathematical models play a crucial role in investigating complex biological systems, enabling a comprehensive understanding of interactions among various components and facilitating in silico testing of intervention strategies. Alzheimer's disease (AD) is characterized by multifactorial causes and intricate interactions among biological entities, necessitating a personalized approach due to the lack of effective treatments. Therefore, mathematical models offer promise as indispensable tools in combating AD. However, existing models in this emerging field often suffer from limitations such as inadequate validation or a narrow focus on single proteins or pathways. Methods: In this paper, we present a multiscale mathematical model that describes the progression of AD through a system of 19 ordinary differential equations. The equations describe the evolution of proteins (nanoscale), cell populations (microscale), and organ-level structures (macroscale) over a 50-year lifespan, as they relate to amyloid and tau accumulation, inflammation, and neuronal death. Results: Distinguishing our model is a robust foundation in biological principles, ensuring improved justification for the included equations, and rigorous parameter justification derived from published experimental literature. Conclusion: This model represents an essential initial step toward constructing a predictive framework, which holds significant potential for identifying effective therapeutic targets in the fight against AD.

Predicting cognitive decline in a low-dimensional representation of brain morphology.

Identifying early signs of neurodegeneration due to Alzheimer's disease (AD) is a necessary first step towards preventing cognitive decline. Individual cortical thickness measures, available after processing anatomical magnetic resonance imaging (MRI), are sensitive markers of neurodegeneration. However, normal aging cortical decline and high inter-individual variability complicate the comparison and statistical determination of the impact of AD-related neurodegeneration on trajectories. In this paper, we computed trajectories in a 2D representation of a 62-dimensional manifold of individual cortical thickness measures. To compute this representation, we used a novel, nonlinear dimension reduction algorithm called Uniform Manifold Approximation and Projection (UMAP). We trained two embeddings, one on cortical thickness measurements of 6237 cognitively healthy participants aged 18-100 years old and the other on 233 mild cognitively impaired (MCI) and AD participants from the longitudinal database, the Alzheimer's Disease Neuroimaging Initiative database (ADNI). Each participant had multiple visits ([Formula: see text]), one year apart. The first embedding's principal axis was shown to be positively associated ([Formula: see text]) with participants' age. Data from ADNI is projected into these 2D spaces. After clustering the data, average trajectories between clusters were shown to be significantly different between MCI and AD subjects. Moreover, some clusters and trajectories between clusters were more prone to host AD subjects. This study was able to differentiate AD and MCI subjects based on their trajectory in a 2D space with an AUC of 0.80 with 10-fold cross-validation.

Efficacy of synaptic inhibition depends on multiple, dynamically interacting mechanisms implicated in chloride homeostasis.

Chloride homeostasis is a critical determinant of the strength and robustness of inhibition mediated by GABA(A) receptors (GABA(A)Rs). The impact of changes in steady state Cl(-) gradient is relatively straightforward to understand, but how dynamic interplay between Cl(-) influx, diffusion, extrusion and interaction with other ion species affects synaptic signaling remains uncertain. Here we used electrodiffusion modeling to investigate the nonlinear interactions between these processes. Results demonstrate that diffusion is crucial for redistributing intracellular Cl(-) load on a fast time scale, whereas Cl(-)extrusion controls steady state levels. Interaction between diffusion and extrusion can result in a somato-dendritic Cl(-) gradient even when KCC2 is distributed uniformly across the cell. Reducing KCC2 activity led to decreased efficacy of GABA(A)R-mediated inhibition, but increasing GABA(A)R input failed to fully compensate for this form of disinhibition because of activity-dependent accumulation of Cl(-). Furthermore, if spiking persisted despite the presence of GABA(A)R input, Cl(-) accumulation became accelerated because of the large Cl(-) driving force that occurs during spikes. The resulting positive feedback loop caused catastrophic failure of inhibition. Simulations also revealed other feedback loops, such as competition between Cl(-) and pH regulation. Several model predictions were tested and confirmed by [Cl(-)](i) imaging experiments. Our study has thus uncovered how Cl(-) regulation depends on a multiplicity of dynamically interacting mechanisms. Furthermore, the model revealed that enhancing KCC2 activity beyond normal levels did not negatively impact firing frequency or cause overt extracellular K(-) accumulation, demonstrating that enhancing KCC2 activity is a valid strategy for therapeutic intervention.

Computational modeling of trans-synaptic nanocolumns, a modulator of synaptic transmission.

Understanding synaptic transmission is of crucial importance in neuroscience. The spatial organization of receptors, vesicle release properties and neurotransmitter molecule diffusion can strongly influence features of synaptic currents. Newly discovered structures coined trans-synaptic nanocolumns were shown to align presynaptic vesicles release sites and postsynaptic receptors. However, how these structures, spanning a few tens of nanometers, shape synaptic signaling remains little understood. Given the difficulty to probe submicroscopic structures experimentally, computer modeling is a useful approach to investigate the possible functional impacts and role of nanocolumns. In our in silico model, as has been experimentally observed, a nanocolumn is characterized by a tight distribution of postsynaptic receptors aligned with the presynaptic vesicle release site and by the presence of trans-synaptic molecules which can modulate neurotransmitter molecule diffusion. In this work, we found that nanocolumns can play an important role in reinforcing synaptic current mostly when the presynaptic vesicle contains a small number of neurotransmitter molecules. Our work proposes a new methodology to investigate in silico how the existence of trans-synaptic nanocolumns, the nanometric organization of the synapse and the lateral diffusion of receptors shape the features of the synaptic current such as its amplitude and kinetics.

Non-invasive neuromodulation for tinnitus: A meta-analysis and modeling studies.

BACKGROUND: Patients with tinnitus often have poor quality of life, as well as severe anxiety and depression. New approaches to treat tinnitus are needed. OBJECTIVE: Evaluate the effects of non-invasive neuromodulation on tinnitus through a metaanalysis and modeling study. The main hypothesis was that real as compared to sham neuromodulation that decreases tinnitus will modulate regions in line with the neurobiological models of tinnitus. METHODS AND RESULTS: The systematic review, conducted from Pubmed, Cochrane and PsycINFO databases, showed that active as compared to sham repetitive transcranial magnetic stimulation (rTMS) reduced tinnitus, but active and sham transcranial direct current stimulation did not significantly differ. Further, rTMS over the auditory cortex was the most effective protocol. The modeling results indicate that this rTMS protocol elicited the strongest electric fields in the insula. Also, rTMS was particularly beneficial in women. Finally, the placebo effects were highly variable, highlighting the importance of conducting sham-controlled trials. CONCLUSION: In sum, neuromodulation protocols that target the auditory cortex and the insula may hold clinical potential to treat tinnitus.

Differential chloride homeostasis in the spinal dorsal horn locally shapes synaptic metaplasticity and modality-specific sensitization.

GABAA/glycine-mediated neuronal inhibition critically depends on intracellular chloride (Cl-) concentration which is mainly regulated by the K+-Cl- co-transporter 2 (KCC2) in the adult central nervous system (CNS). KCC2 heterogeneity thus affects information processing across CNS areas. Here, we uncover a gradient in Cl- extrusion capacity across the superficial dorsal horn (SDH) of the spinal cord (laminae I-II: LI-LII), which remains concealed under low Cl- load. Under high Cl- load or heightened synaptic drive, lower Cl- extrusion is unveiled in LI, as expected from the gradient in KCC2 expression found across the SDH. Blocking TrkB receptors increases KCC2 in LI, pointing to differential constitutive TrkB activation across laminae. Higher Cl- lability in LI results in rapidly collapsing inhibition, and a form of activity-dependent synaptic plasticity expressed as a continuous facilitation of excitatory responses. The higher metaplasticity in LI as compared to LII differentially affects sensitization to thermal and mechanical input. Thus, inconspicuous heterogeneity of Cl- extrusion across laminae critically shapes plasticity for selective nociceptive modalities.

Enhancing neuronal chloride extrusion rescues α2/α3 GABAA-mediated analgesia in neuropathic pain.

Spinal disinhibition has been hypothesized to underlie pain hypersensitivity in neuropathic pain. Apparently contradictory mechanisms have been reported, raising questions on the best target to produce analgesia. Here, we show that nerve injury is associated with a reduction in the number of inhibitory synapses in the spinal dorsal horn. Paradoxically, this is accompanied by a BDNF-TrkB-mediated upregulation of synaptic GABAARs and by an α1-to-α2GABAAR subunit switch, providing a mechanistic rationale for the analgesic action of the α2,3GABAAR benzodiazepine-site ligand L838,417 after nerve injury. Yet, we demonstrate that impaired Cl- extrusion underlies the failure of L838,417 to induce analgesia at high doses due to a resulting collapse in Cl- gradient, dramatically limiting the benzodiazepine therapeutic window. In turn, enhancing KCC2 activity not only potentiated L838,417-induced analgesia, it rescued its analgesic potential at high doses, revealing a novel strategy for analgesia in pathological pain, by combined targeting of the appropriate GABAAR-subtypes and restoring Cl- homeostasis.

Improved Simulation of Electrodiffusion in the Node of Ranvier by Mesh Adaptation.

In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.

Chloride Regulation: A Dynamic Equilibrium Crucial for Synaptic Inhibition.

Fast synaptic inhibition relies on tight regulation of intracellular Cl(-). Chloride dysregulation is implicated in several neurological and psychiatric disorders. Beyond mere disinhibition, the consequences of Cl(-) dysregulation are multifaceted and best understood in terms of a dynamical system involving complex interactions between multiple processes operating on many spatiotemporal scales. This dynamical perspective helps explain many unintuitive manifestations of Cl(-) dysregulation. Here we discuss how taking into account dynamical regulation of intracellular Cl(-) is important for understanding how synaptic inhibition fails, how to best detect that failure, why Cl(-) regulation is energetically so expensive, and the overall consequences for therapeutics.

Mild KCC2 Hypofunction Causes Inconspicuous Chloride Dysregulation that Degrades Neural Coding.

Disinhibition caused by Cl(-) dysregulation is implicated in several neurological disorders. This form of disinhibition, which stems primarily from impaired Cl(-) extrusion through the co-transporter KCC2, is typically identified by a depolarizing shift in GABA reversal potential (E GABA). Here we show, using computer simulations, that intracellular [Cl(-)] exhibits exaggerated fluctuations during transient Cl(-) loads and recovers more slowly to baseline when KCC2 level is even modestly reduced. Using information theory and signal detection theory, we show that increased Cl(-) lability and settling time degrade neural coding. Importantly, these deleterious effects manifest after less KCC2 reduction than needed to produce the gross changes in E GABA required for detection by most experiments, which assess KCC2 function under weak Cl(-) load conditions. By demonstrating the existence and functional consequences of "occult" Cl(-) dysregulation, these results suggest that modest KCC2 hypofunction plays a greater role in neurological disorders than previously believed.

Morphine hyperalgesia gated through microglia-mediated disruption of neuronal Cl⁻ homeostasis.

A major unresolved issue in treating pain is the paradoxical hyperalgesia produced by the gold-standard analgesic morphine and other opiates. We found that hyperalgesia-inducing treatment with morphine resulted in downregulation of the K(+)-Cl(-) co-transporter KCC2, impairing Cl(-) homeostasis in rat spinal lamina l neurons. Restoring the anion equilibrium potential reversed the morphine-induced hyperalgesia without affecting tolerance. The hyperalgesia was also reversed by ablating spinal microglia. Morphine hyperalgesia, but not tolerance, required µ opioid receptor-dependent expression of P2X4 receptors (P2X4Rs) in microglia and µ-independent gating of the release of brain-derived neurotrophic factor (BDNF) by P2X4Rs. Blocking BDNF-TrkB signaling preserved Cl(-) homeostasis and reversed the hyperalgesia. Gene-targeted mice in which Bdnf was deleted from microglia did not develop hyperalgesia to morphine. However, neither morphine antinociception nor tolerance was affected in these mice. Our findings dissociate morphine-induced hyperalgesia from tolerance and suggest the microglia-to-neuron P2X4-BDNF-KCC2 pathway as a therapeutic target for preventing hyperalgesia without affecting morphine analgesia.