Neuroscience

Fitting bifurcation structure, not voltage traces: A biophysically inspired derivation of reduced neuron models exemplified by potassium dynamics

Louisiane Lemaire
Mahraz Behbood
Jan-Hendrik Schleimer
Susanne Schreiber author has email address

Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Berlin, Germany
Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Berlin, Germany
Inria Branch of the University of Montpellier, Montpellier, France

https://doi.org/10.7554/eLife.108270.1

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Figures and data

Schematic representation of the two models: the target model (a) and its corresponding phenomenological version (b).
(a) Conductance-based model of excitability class I with dynamic extracellular potassium concentration, equipped with the usual Na⁺ and K⁺ voltage-gated channels and with an electrogenic Na⁺/K⁺ ATPase (diagram reproduced from [7]). (b) Quadratic integrate-and-fire (QIF) model with dynamic extracellular potassium concentration, equipped with an electrogenic Na⁺/K⁺ ATPase. Each spike (i.e., when the voltage v exceeds a threshold v_th) triggers two reset rules: as in the classical QIF model the voltage is reset to a value v_r, and the extracellular potassium concentration is additionally incremented by a small amount Δ_K.

Matching the bifurcation structure of the target model with the QIF model, at fixed values of .
(a)-(b) Bifurcation diagrams of the fast subsystem of the Wang-Buzsáki model (Eqs. (4a) to (4c)) as a function of the applied current, featuring SNIC spike onset when (a) and homoclinic onset when (b). (c)-(d) Bifurcation diagrams of the model defined in Eqs. (1a) and (1b), with parameter values chosen to mimic the diagrams in (a) or (b), respectively. Limit cycles are shown in purple and fixed points in black, with solid lines for stable branches and dashed lines for unstable branches.

Matching the bifurcation structure of the target model with the QIF model, at fixed values of .
(a)-(b) Bifurcation diagrams of the fast subsystem of the Wang-Buzsáki model (Eqs. (4a) to (4c)) as a function of the applied current, featuring SNIC spike onset when (a) and homoclinic onset when (b). (c)-(d) Bifurcation diagrams of the model defined in Eqs. (1a) and (1b), with parameter values chosen to mimic the diagrams in (a) or (b), respectively. Limit cycles are shown in purple and fixed points in black, with solid lines for stable branches and dashed lines for unstable branches.

Bifurcation structure and frequency landscape near the SNL bifurcation.
(a) Two-parameter bifurcation diagram versus applied current I_app of the fast subsystem of the Wang-Buzsáki model (Eqs. (4a) to (4c)) and frequency of the limit cycles. (b) Same as (a), for the QIF model (Eqs. (1a) and (1b)). Note that in this model, the firing frequency goes to infinity as v_r approaches v_th. Here we only show the frequencies up to a certain distance between those parameters.

Bifurcation structure and frequency landscape near the SNL bifurcation.
(a) Two-parameter bifurcation diagram versus applied current I_app of the fast subsystem of the Wang-Buzsáki model (Eqs. (4a) to (4c)) and frequency of the limit cycles. (b) Same as (a), for the QIF model (Eqs. (1a) and (1b)). Note that in this model, the firing frequency goes to infinity as v_r approaches v_th. Here we only show the frequencies up to a certain distance between those parameters.

Parameters of the QIF model (Eqs. (1a) and (1b)) depending on the potassium concentration.
(a-d) Derivation of the parameters I_SN,0, v_SN, a and v_r. (e) Bifurcation diagram of the fast subsystem of the target model (Eqs. (4a) to (4c)) with respect to , when I_app = I_app,SNL and I_pump = 0 µA cm⁻².

Parameters of the QIF model (Eqs. (1a) and (1b)) depending on the potassium concentration.
(a-d) Derivation of the parameters I_SN,0, v_SN, a and v_r. (e) Bifurcation diagram of the fast subsystem of the target model (Eqs. (4a) to (4c)) with respect to , when I_app = I_app,SNL and I_pump = 0 µA cm⁻².

Parameter values of the QIF model (2) to match the bifurcation structure of the Wang-Buzsáki model (4).
The values of the parameters γ, I_pump,max, k and K_eq are unchanged compared to Table III.

Three scenarios of activity-mediated switches in firing regimes that were reported in class I conductance-based models.
(a-c) Voltage traces, in the case of the Wang-Buzsáki model. (d) Schematic representation of trajectories in the versus applied current bifurcation diagram, through regions associated with various activity regimes.

Three scenarios of activity-mediated switches in firing regimes that were reported in class I conductance-based models.
(a-c) Voltage traces, in the case of the Wang-Buzsáki model. (d) Schematic representation of trajectories in the versus applied current bifurcation diagram, through regions associated with various activity regimes.

Fold/homoclinic bursting scenario (deterministic input: I_app = 0.25 µA cm⁻²).
For the Wang-Buzsáki model: (a) Bursting trajectory shown on the 2-parameter bifurcation diagram [K⁺]_o versus I_app of the fast subsystem. (b) Voltage and [K⁺]_o time traces. (c) Enlargement of the burst highlighted in panel (b). (d) Bursting trajectory superimposed onto the 1-parameter bifurcation diagram with respect to [K⁺]_o, zooming in on lower voltage values for better visibility. (e) Time derivative of the averaged slow subsystem. (f-j) Same as (a-e), for the integrate-and-fire version.

Stochastic bursting scenario (noisy input, with mean I_app = 0.3 µA cm⁻²).
For the Wang-Buzsáki model: (a) Trajectory shown on the 2-parameter bifurcation diagram [K⁺]_o versus I_app of the fast subsystem. (b) Voltage, [K⁺]_o and applied current time traces. (c) Enlargement of the burst highlighted in panel (b). (d) Part of the trajectory corresponding to the burst highlighted in panels (b) and (f), superimposed onto the 1-parameter bifurcation diagram with respect to [K⁺]_o. (e-h) same as (a-d), for the integrate-and-fire version.

Depolarization block scenario (deterministic input: I_app = 0.3 µA cm⁻²).
For the Wang-Buzsáki model: (a) Trajectory shown on the 2-parameter bifurcation diagram [K⁺]_o versus I_app of the fast subsystem. (b) Voltage and [K⁺]_o time traces. (c) Enlargement of the transition to depolarization block (b). (d) Part of the trajectory highlighted in panels (b) and (f), superimposed onto the 1-parameter bifurcation diagram with respect to [K⁺]_o. (e-h) Same as (a-d), for the integrate-and-fire version.

Effect of on the phase response curve and ability to synchronize, near spike onset .
(a) iPRCs for the Wang-Buzsáki model (direct method). (b) Spike voltage traces for the Wang-Buzsáki model. (c) iPRCs for the integrate-and-fire model (analytically, see “Methods”). (d) Spike voltage traces for the integrate-and-fire model. (e) Period of the limit cycles. (f) Locking range, assuming delta coupling with a voltage perturbation of 0.15 µV. For the computation of the coupling function, pulses were normalised following Weerdmeester et al. [23].

Effect of on the phase response curve and ability to synchronize, near spike onset .
(a) iPRCs for the Wang-Buzsáki model (direct method). (b) Spike voltage traces for the Wang-Buzsáki model. (c) iPRCs for the integrate-and-fire model (analytically, see “Methods”). (d) Spike voltage traces for the integrate-and-fire model. (e) Period of the limit cycles. (f) Locking range, assuming delta coupling with a voltage perturbation of 0.15 µV. For the computation of the coupling function, pulses were normalised following Weerdmeester et al. [23].

Response of a network of 100 neurons to an external potassium wave.
We considered an all-to-all connectivity scenario, with weak inhibitory pulse-coupling (δ_v = −0.000 15 mV). We used a noisy input (see “Methods”), common to all neurons, with mean I_app = 0.25 µA cm⁻². (a) Incoming potassium wave, common to all neurons. (b) 2-parameter bifurcation diagram versus I_app of the fast subsystem, for the Wang-Buzsáki model. The crosses indicate the initial and final potassium concentrations. (c-d) Raster plot for 10 among the 100 neurons, during the third and 13^th seconds, respectively. (e-g) Same as (b-d), for the QIF version of the model.

Response of a network of 100 neurons to an external potassium wave.
We considered an all-to-all connectivity scenario, with weak inhibitory pulse-coupling (δ_v = −0.000 15 mV). We used a noisy input (see “Methods”), common to all neurons, with mean I_app = 0.25 µA cm⁻². (a) Incoming potassium wave, common to all neurons. (b) 2-parameter bifurcation diagram versus I_app of the fast subsystem, for the Wang-Buzsáki model. The crosses indicate the initial and final potassium concentrations. (c-d) Raster plot for 10 among the 100 neurons, during the third and 13^th seconds, respectively. (e-g) Same as (b-d), for the QIF version of the model.

Variables of the extended Wang-Buzsáki model [7] given in Eq. (4).

Parameter values for the extended Wang-Buzsáki model [7] given in Eq. (4).

Bifurcation diagrams with respect to [K⁺]_o and frequency of the limit cycles when I_app = 0.25 µA cm⁻², for the Wang-Buzsáki (a,b) and integrate-and-fire (c,d) models. (e-h) Same as (a-d), when I_app = 0.3 µA cm⁻².

Tonic firing scenario, when adding a diffusion term to the potassium dynamics, with diffusion coefficient D = 0.4 Hz and bath concentration K_bath = 5 mM.
The initial conditions and other parameter values are unchanged compared to the fold/homoclinic bursting scenario. For the Wang-Buzsáki model: (a) Bursting trajectory shown on the 2-parameter bifurcation diagram [K⁺]_o versus I_app of the fast subsystem. (b) Voltage and [K⁺]_o time traces. (c) Enlargement of (b). (d) Trajectory superimposed onto the 1-parameter bifurcation diagram with respect to [K⁺]_o. (e) Time derivative of the averaged slow subsystem. (f-j) Same as (a-e), for the integrate-and-fire version.

Tonic firing scenario, when adding a diffusion term to the potassium dynamics, with diffusion coefficient D = 0.4 Hz and bath concentration K_bath = 5 mM.
The initial conditions and other parameter values are unchanged compared to the fold/homoclinic bursting scenario. For the Wang-Buzsáki model: (a) Bursting trajectory shown on the 2-parameter bifurcation diagram [K⁺]_o versus I_app of the fast subsystem. (b) Voltage and [K⁺]_o time traces. (c) Enlargement of (b). (d) Trajectory superimposed onto the 1-parameter bifurcation diagram with respect to [K⁺]_o. (e) Time derivative of the averaged slow subsystem. (f-j) Same as (a-e), for the integrate-and-fire version.

Schematic representation of the relation between the spike voltage trace and iPRC in the QIF model, a one-dimensional model.
(a-b) iPRC near spike onset and spike voltage trace when (physiological). The iPRC is largest when the derivative of the solution is smallest. (c-d) iPRC near spike onset (same ) and spike voltage trace at the SNL , i.e., when the reset is at the inflection point.

Schematic representation of the relation between the spike voltage trace and iPRC in the QIF model, a one-dimensional model.
(a-b) iPRC near spike onset and spike voltage trace when (physiological). The iPRC is largest when the derivative of the solution is smallest. (c-d) iPRC near spike onset (same ) and spike voltage trace at the SNL , i.e., when the reset is at the inflection point.

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