Biophysically inspired mean-field model of neuronal populations driven by ion-exchange mechanisms
- Aix Marseille University, Institut de Neurosciences des Systèmes, UMR 1106, INSERM, France
- MathNeuro, INRIA, University of Montpellier, France
- MCENteam, Basque Center for Applied Mathematics (BCAM), Spain
- Faculty of Medicine and Health Technology, Tampere University, Finland
Figures
Figure 1
Biophysically inspired neural mass model.
Schematic diagram of the ion channel mechanism in extracellular and intracellular space in the brain. A biophysical model of a single neuron consists of three compartments (left panel): the intracellular space (ICS; in red), the extracellular space (ECS; in dark gray), and the external bath (EB; in light gray). The ion exchange across the cellular space occurs through the ion channels: Na+ gets inside the ICS (yellow channel), K+ gets out (green channel), the flow of Cl− can be bidirectional (purple channel); for the pump (blue), Na+ gets out and K+ gets into the ICS. A population of interacting neurons sharing the same concentration forms a local neural mass (middle panel), for which we model the mean-field equations in this work. Brain network model (right panel) with the activity of each brain region represented by neural masses.
Figure 2 with 1 supplement
Single Hodgkin–Huxley-type neuron model.
(a) Different patterns of electrophysiological activities previously identified in Depannemaecker et al., 2022b are also reproduced in our parameter setting by varying the potassium concentration in the external bath . The membrane potential V is measured in mV and the ion concentrations in mmol/m3. (b) Phase space trajectory of the seizure-like event simulation (). Fast oscillations occur in a fast subsystem identified by the membrane potential V and the gating variable n. The oscillations of the slow subsystem, here captured by the potassium concentration in the extracellular space , enable the transition to bursting. (c) Fixing the value of the state variables , and as constants, the membrane potential equation resembles a cubic function for different values of . We can model this function as a step-wise quadratic approximation, corresponding to two parabolas with vertices at coordinates and and curvature and , respectively (d). The two parabolas meet at an intersection point where the membrane potential equation changes curvature. (e) At each time, we assume that the membrane potential of a neuronal population is distributed according to a Lorentzian centered at and with width , for each value of the excitability η (Lorentzian Ansatz). In the case depicted, the cubic function meets the zero for , the neuronal population is described by the Lorentzian distribution in blue in the steady-state solution, and the neuronal dynamics is governed by the positive parabola according to the continuity equation. In the case where the derivative of the membrane potential crosses zero for (e.g., if the cubic function is shifted up by adding a constant current to the membrane potential derivative), the population is described by the red distribution in the steady state, and the continuity equation is governed by the negative parabola equation. Cases where the cubic function meets the zero in more than one point are not well described by this approximation, see Steady-state solution and Lorentzian Ansatz section.
Figure 2—figure supplement 1
Quadratic approximation of nullcline geometry.
(a) Neuronal network simulations of all-to-all coupled Hodgkin–Huxley-type neurons can display bimodal distribution of the membrane potential when a subpopulation transitions from sub- to supra-threshold activity. (b) By changing the values of the parabola coefficients, which describe the nullcline geometry of the model, the results of the mean-field simulation change qualitatively. We show this parameters’ effect by plotting the power spectral density for several values of these parameters (here we used ). We can observe that for all the parameters, frequency shifts and sudden state transitions occur.
Figure 3 with 1 supplement
Mean-field model versus neuronal network across dynamical regimes.
(a) Example raster plot of a population of all-to-all coupled HH-type neurons displaying sub- and supra-threshold dynamics that can be well described by a Lorentzian distribution. (b) For several values we simulated the activity of a population of all-to-all coupled HH-type neurons across dynamical regimes (parameters in Supplementary file 1). We compare the mean membrane potential and external potassium of such population (in blue) with the results obtained using the mean-field model equations (in green). To properly match the slow timescale of the population, we defined an effective value of the potassium concentration in the bath , represented in panel (c) (the dashed line represents the identity).
Figure 3—figure supplement 1
Quantifying the error of the first moment approximation.
(a) For several values corresponding to different dynamical regimes we simulated the activity of a population of N = 3000 all-to-all coupled HH-type neurons (parameters same as Figure 3a in Supplementary file 1) and classified them in three states: quiscent (blue), bursting (orange), and depolarized (crimson red) based on their individual gating variable (n) values. Along with that, using the right-hand axis, the population mean of the gating variable is plotted in green. Errors (black dotted) have been calculated by calculating which fractions of neurons are not following the state of the mean gating variable. (b) The error percentage (neurons not following the mean behavior) is plotted against several values.
Figure 4
Comparison of numerical results and in vitro experiments.
(a) Neural mass model showing slow periodic fluctuations in extracellular potassium concentration (green) with voltage bursts (blue) riding on top. (b) Network simulation of coupled HH-type neurons exhibiting a similar bursting pattern at a shorter timescale (due to rescaled parameters). (c) In vitro recording showing LFP; blue, AC-coupled and extracellular potassium concentration (green), exhibiting slow periodic fluctuations and bursting activity. (d) In vitro trace showing more complex dynamics – after a shift to a high-potassium state (marked by arrows), the burst frequency slows down progressively. (e) Simulation from the mean-field model showing an emergent bursting regime with isolated events in the up state, not seen at the single-neuron level.
Figure 5
Fast subsystem bifurcation diagram in two parameters.
(a) The bifurcation diagram shows the behavior of the fast subsystem of system (Equation 38) to the slow variables and , which are parameters in the fast-subsystem limit. Each curve represents parameter values for which specific bifurcations occur, obtained through a numerical continuation algorithm. Saddle Node (SN) bifurcations are shown in blue, Saddle Homoclinic (SH) bifurcations in black, Fold Limit Cycle (FLC) bifurcations in green, and Hopf bifurcations in red. Colored dots indicate codimension-2 points. In particular, a Bogdanov–Takens (BT) point and a Saddle-Node-Loop (SNL) point also identified in the single-neuron model, and epileptor model. Note that this diagram is quite involved and we do not claim that the present version is complete, however it is representative of the complexity of the bifurcation structure of the mean-field model. (b1, b2) Comparison of the fast subsystem bifurcation diagrams for the single-neuron model and the neural mass model. The left side shows a zoomed-in version of the diagram from panel (a) to facilitate comparison with the single-neuron model diagram on the right. Both diagrams use the same color coding as described above. This comparison highlights the similarities and differences in the bifurcation structures of these two models and indicates emergent structures due to interactions within the network.
Figure 6
Network simulation of structurally connected neural mass models and propagation of pathological bursts.
(a) Structural connectivity for six all-to-all connected nodes A, B, C, D, E, and F with random weight allocation. Each node is described by a neural mass model derived as the mean-field approximation of a large population of Hodgkin–Huxley (HH)-type neurons. (b) When decoupled (global coupling ), all the nodes operate in a ‘healthy’ regime with the potassium concentration in their bath set to low values , except for node D which is tuned into a pathological regime characterized by the spontaneous presence of bursts. (c) When the global coupling is sufficiently increased, the pathological value of in node D generates bursts that diffuse through the connectome mimicking the spreading of a seizure.
Tables
Table 1
List of parameters and their values used for the single-neuron simulation.
| Parameter | Symbol | Value |
|---|---|---|
| Membrane capacitance | 1nF | |
| Gating time constant | 4ms | |
| Chloride conductance | 7.5 nS | |
| Maximal potassium conductance | 22 nS | |
| Maximal sodium conductance | 40 nS | |
| Potassium leak conductance | 0.12 nS | |
| Sodium leak conductance | 0.02 nS | |
| Intracellular volume | 2160 μm3 | |
| Extracellular volume | 720 μm3 | |
| Intra-/extracellular volume ratio | 3 | |
| Conversion factor | γ | 0.04 mol/C |
| Diffusion rate | ε | 0.001 mHz |
| Maximal Na/K pump current | ρ | 250 pA |
| Initial concentration of extracellular K | 4.8 mmol/m3 | |
| Initial concentration of intracellular K | 130 mmol/m3 | |
| Initial concentration of extracellular Na | 138 mmol/m3 | |
| Initial concentration of intracellular Na | 16 mmol/m3 | |
| Initial concentration of extracellular Cl | 112 mmol/m3 | |
| Initial concentration of intracellular Cl | 5 mmol/m3 |
Additional files
-
Supplementary file 1
List of parameters’ values used for each simulation in the manuscript.
If not present, the parameter is not applicable for that case, or different values have been used, as specified in the corresponding caption.
- https://cdn.elifesciences.org/articles/104249/elife-104249-supp1-v1.docx
-
MDAR checklist
- https://cdn.elifesciences.org/articles/104249/elife-104249-mdarchecklist1-v1.pdf
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Biophysically inspired mean-field model of neuronal populations driven by ion-exchange mechanisms
eLife 14:RP104249.
https://doi.org/10.7554/eLife.104249.3
