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 [13]). (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 vth) triggers two reset rules: as in the classical QIF model the voltage is reset to a value vr, and the extracellular potassium concentration is additionally incremented by a small amount ΔK.
Matching the bifurcation structure of the target model with the QIF model, at fixed values of [K+]o.
(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 [K+]o = 5 mM (a) and homoclinic onset when [K+]o = 12 mM (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 dotted lines for unstable branches.
Bifurcation structure and frequency landscape near the SNL bifurcation.
(a) Two-parameter bifurcation diagram [K+]o versus applied current Iapp 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 vr approaches vth. 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 ISN,0, vSN, a and vr. Note that the increase of the reset voltage vr with [K+]o reflects the increase of the potassium reversal potential EK in the conductance-based model (panel (d)). (e) Bifurcation diagram of the fast subsystem of the target model (Eqs. (4a) to (4c)) with respect to [K+]o, when Iapp = Iapp,SNL (i.e., the value of applied current at which the SNL bifurcation occurs) and Ipump = 0 µA cm−2. In this panel, we have reproduced the linear fit of the reset voltage vr shown in panel (d), for visual comparison with the minimal voltage values of the limit cycles.
Steps to derive a quadratic integrate-and-fire model with dependence on a biophysical parameter, by fitting the bifurcation structure of a class I conductance-based neuron model near an SNL bifurcation induced by this parameter.
Note that, in this paper, we performed these reduction steps in the absence of pump current in the original model, and added this current a posteriori to the reduced model.
Parameter values of the QIF model (3) to match the bifurcation structure of the fast subsystem of the Wang-Buzsáki model (4) with respect to [K+]o.
The values of the parameters γ, Ipump,max, k and Keq are unchanged compared to Table IV.
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 [K+]o versus applied current bifurcation diagram through regions associated with various activity regimes.
For the Wang-Buzsáki model: (a) Bursting trajectory shown on the 2-parameter bifurcation diagram [K+]o versus Iapp 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 of the fast subsystem with respect to [K+]o, zooming in on lower voltage values for better visibility. (e) Time derivative of the averaged and standard slow subsystems. For the quadratic integrate-and-fire model: (f-j) Same as (a-e).
Stochastic bursting scenario (noisy input, with mean Iapp = 0.3 µA cm−2).
For the Wang-Buzsáki model: (a) Trajectory shown on the 2-parameter bifurcation diagram [K+]o versus Iapp 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 of the fast subsystem with respect to [K+]o. For the quadratic integrate-and-fire model: (e-h) Same as (a-d).
For the Wang-Buzsáki model: (a) Trajectory shown on the 2-parameter bifurcation diagram [K+]o versus Iapp of the fast subsystem. (b) Voltage and [K+]o time traces. (c) Enlargement of the transition to depolarization block shown in panel (b). (d) Part of the trajectory highlighted in panels (b) and (c), superimposed onto the 1-parameter bifurcation diagram of the fast subsystem with respect to [K+]o. For the quadratic integrate-and-fire model: (e-h) Same as (a-d).
Effect of [K+]o on the phase response curve and ability to synchronize near spike onset .
(a) iPRCs for the Wang-Buzsáki model (direct method). Note that the curve for [K+]o = 7.21 mM lies beneath the curve for [K+]o = 5.21 mM; the same holds for panel (c). (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 pulse coupling with a voltage perturbation of 0.15 µV. For the computation of the coupling function, pulses were normalized following Weerdmeester et al. [31].
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 (δv = −0.000 15 mV). We used a noisy input (see “Methods”), common to all neurons, with mean Iapp = 0.25 µA cm−2. (a) Incoming potassium wave, common to all neurons. (b) 2-parameter bifurcation diagram [K+]o versus Iapp 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 13th seconds, respectively. (e-g) Same as (b-d), for the QIF version of the model.
Variables of the extended Wang-Buzsáki model [13] given in Eq. (4).
Parameter values for the extended Wang-Buzsáki model [13] given in Eq. (4).
Bifurcation diagram of the classical QIF model, defined by: (with a > 0); if V > Vth, then V ← Vr.
When b is nega tive, the system has a stable fixed point at and an unstable fixed point at . They disappear via a saddle-node bifurcation at b = 0. When the reset voltage Vr is negative, periodic solutions emerge via a SNIC bifurcation at b = 0. When Vr is positive, periodic solutions are instead created via a homoclinic bifurcation, at . In this case, for , the system exhibits bistability between the stable limit cycle and the stable fixed point. The transition from SNIC spike onset to homoclinic spike onset is mediated by a SNL bifurcation at Vr = b = 0 (i.e., when the voltage is reset to the saddle-node point), a codimension-2 bifurcation which acts here as an organizing center. Adapted from Figure 8.3 in [8].
Illustration of the reduction procedure, here applied to derive a QIF model that captures dependence on temperature instead of potassium concentration, motivated by Hesse et al. [9].
The initial conductance-based model is the Wang–Buzsáki model as in [9]: [K+]o is constant, there is no pump current, and temperature dependence is introduced via the Nernst equation and Q10 coefficients. The resulting QIF model reproduces the transition induced by temperature rise reported in Figure 3 of [9], characterized by intermittently interrupted firing in response to noisy input. (a-e) QIF parameters derivation: ISN (a), vSN (b), a (c), vr (d) and vth (e).(f-j) Two-parameter bifurcation diagram temperature T versus applied current Iapp, for the initial (f) and QIF (j) models. (g-i) Voltage response to a noisy input current Inoisy (Eq. (9) in “Methods”, with τη = 0.01 s and σ = 0.1 µA cm−2) at low (g) or high (h) temperature, in the initial model. The mean applied current Iapp is chosen smaller in the HOM case (high temperature) than in the SNIC case, to obtain a similar number of spikes per second (see red crosses in panels (f) and (j)). The time course of η is shown in panel (i). (k-m) Same as (g-i), in the QIF model.
Bifurcation diagrams with respect to [K+]o and frequency of the limit cycles when Iapp = 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 Iapp = 0.3 µA cm−2.
Tonic firing scenario, when adding a diffusion term Idiff = D([K+]o − Kbath) to the potassium dynamics, with diffusion coefficient D = 0.4 Hz and bath concentration Kbath = 5 mM.
The initial conditions and other parameter values are unchanged compared to the fold/homoclinic bursting scenario. For the Wang-Buzsáki model: (a) Trajectory shown on the 2-parameter bifurcation diagram [K+]o versus Iapp 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 [K+]o = 5.21 mM (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 ([K+]o = 9.21 mM), i.e., when the reset is at the inflection point.
