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MCurrent Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains
The Journal of Mathematical Neuroscience volume 8, Article number: 13 (2018)
Abstract
Theta (4–8 Hz) and gamma (30–80 Hz) rhythms in the brain are commonly associated with memory and learning (Kahana in J Neurosci 26:1669–1672, 2006; Quilichini et al. in J Neurosci 30:11128–11142, 2010). The precision of cofiring between neurons and incoming inputs is critical in these cognitive functions. We consider an inhibitory neuron model with Mcurrent under forcing from gamma pulses and a sinusoidal current of theta frequency. The Mcurrent has a long time constant (∼90 ms) and it has been shown to generate resonance at theta frequencies (Hutcheon and Yarom in Trends Neurosci 23:216–222, 2000; Hu et al. in J Physiol 545:783–805, 2002). We have found that this slow Mcurrent contributes to the precise cofiring between the network and fast gamma pulses in the presence of a slow sinusoidal forcing. The Mcurrent expands the phaselocking frequency range of the network, counteracts the slow theta forcing, and admits bistability in some parameter range. The effects of the Mcurrent balancing the theta forcing are reduced if the sinusoidal current is faster than the theta frequency band. We characterize the dynamical mechanisms underlying the role of the Mcurrent in enabling a network to be entrained to gamma frequency inputs using averaging methods, geometric singular perturbation theory, and bifurcation analysis.
Introduction
Gamma (30–80 Hz) and theta (4–8 Hz) oscillations are prominent rhythms observed in local field potentials recorded from entorhinal–hippocampal circuits [2, 5, 6]. Gamma oscillations often emerge at specific phases of the slower theta oscillations, a phenomenon known as theta–gamma crossfrequency coupling (CFC) [7, 8]. Entorhinal–hippocampal theta–gamma CFC is thought to be critical in the process of memory formation [9,10,11,12]. Computational modeling suggests theta–gamma CFC allows gamma oscillations from different regions to be coordinated temporally, since theta oscillations can be synchronized over larger spatial areas than gamma [13]. Precise coordination of spikes is critical for spiketimedependent plasticity (STDP) thought to underlie mechanisms of learning and memory [14]. Models also suggest the theta component of the theta–gamma coupling may serve either as a phase reference for the alignment of the gamma oscillations [13, 15] or as a mechanism that periodically breaks communication between neuronal ensembles and allows the system to reset [16].
Here, using computational models, we find an additional, nonintuitive function for theta–gamma CFC: theta–gamma CFC can effectively allow one gamma oscillator to precisely regulate the gamma spiking of the theta–gamma targeted oscillator. We show that the underlying mechanism involves the theta timescale of an intrinsic membrane potassium current known as the Mcurrent. This is surprising because the Mcurrent (internal theta timescale) interacts with both external gamma and theta timescales. Even though gammaaminobutyric acid (GABA) synaptic currents are a key ingredient in the generation of gamma oscillations [17], their presence is not enough to secure entrainment by a gamma input in the presence of a theta input in our model. Note that, in this paper, we emphasize the order of spike arrival by using the expression “a cell follows the inputs.” In particular, we only consider the type of entrainment where the output spikes occur after the input spikes, due to its implication in STDP [14].
For this purpose, we consider a small neuronal network consisting of a single cell with an inhibitory autapse, where the cell is provided with an Mcurrent. Such a network can represent the simplest form of a pyramidalinterneuronal network, or PING [18]; the inhibitory autapse then represents the inhibitory feedback to the pyramidal cell. The Mcurrent can exist in either excitatory cells [4] or inhibitory cells [19] in the network. Such a network exists in abundance in the brain [20,21,22], in particular in the hippocampus [23, 24]. In the latter case, it is known that there are both gamma [5] and theta [6, 25] inputs, as we use in our model. Thus, our model can represent hippocampal networks and their inputs, and possibly be generalized to other situations in which the target network receives gamma and theta inputs.
Mechanistically, the Mcurrent plays a dual role in the network. In the subthreshold regime, it interacts with the external theta current, providing homeostasis and stabilizing the subthreshold voltage fluctuations in a frequencydependent manner. In the spiking regime, it interacts with the external gamma pulses. It expands the range of frequencies over which the oscillator can follow. In particular, the Mcurrent allows the oscillator to \(1:1\) phaselock to a rhythm slower than its natural frequency. Moreover, it enables the oscillator to be sparsely entrained by fast inputs (i.e., the oscillator skips some of the input cycles but always closely follows the inputs).
The Mcurrent is able to provide subthreshold homeostasis as long as the external current is in the theta frequency band or slower. We show this by simulating an averaged system and observe that the homeostatic effect gradually wears off as the frequency of external current increases, which results in the loss of entrainment. The mechanism of Mcurrent in the spiking regime is more complex. We first examine the phase plane of a reduced model to show that when the oscillator with Mcurrent receives pulses, the reset point is farther away from the knee of the voltage nullcline (in a vicinity of the slow manifold created by the Mcurrent) than when there is no pulse. This is due to pulsetriggered spikes having a larger amplitude and therefore generating a bigger amount of Mcurrent than autonomous spikes. Hence, after an external pulse triggers a spike, it takes the oscillator longer than its natural interspike interval to generate the next spike, allowing the cell to be able to follow a rhythm slower than its natural frequency. Without the Mcurrent, there is no slow manifold; the interspike intervals are the same with autonomous spikes and triggered spikes. Next, utilizing geometric singular perturbation theory, we study a network of an input cell (Ecell) forcing an oscillator with an inhibitory autapse (Icell). Considering dynamics in the singular limit of the slow timescale, two relevant fold structures exist: the Efold determines the spiking of the Ecell and the Ifold determines the spiking of the Icell. Our results show that the presence of an Mcurrent alters the position of the Ifold such that the singular orbit of limit cycle always reaches the Efold first, allowing the Icell to be driven precisely (i.e., sparsely entrained) by the Ecell (external pulses). While the reduced model tells us that the interspike interval is lengthened due to the position of reset point and the slow manifold created by the Mcurrent, the geometric singular perturbation analysis tells us that the fold is also further away from the trajectory when the Mcurrent is stronger.
In addition, we present a special case, where the oscillator with Mcurrent can be entrained by an arbitrarily slow rhythm, due to bistability. We also discuss the similarities and differences between an Mcurrent and an hcurrent in Sect. 6, as hcurrent is also a resonant current and it operates at a similar timescale as the Mcurrent [4].
Mathematical Model
The objective is to study how an Mcurrent in a small network improves the precision of cofiring between gamma inputs and the target network in the presence of a current of theta frequency. We consider a Hodgkin–Huxleylike model for a cell with an autapse and the Mcurrent (referred to as the Icell in the rest of the paper), which can also be thought as a PING network (by regarding the cell as the pyramidal cell in PING and the inhibitory autapse as the inhibitory feedback from interneurons to pyramidal cells). Theta input is modeled as a slowly changing (sinusoidal) oscillation (4 Hz) representing incoming activity from a large ensemble of neurons. In contrast, we hypothesize that gamma inputs are more synchronous, needed to mark the precise moment at which they arrive. To this end, we make the gamma input very sharp over a short time frame (less than 1 ms) to resemble excitatory input from spike trains (see red trace in the top panel of Fig. 1).
The differential equations describing the neuron are as follows:
where \(I_{L}\), \(I_{K}\), \(I_{\mathit {Na}}\), \(I_{\mathit {GABA}_{A}}\), and \(I_{M}\) are the leak current, potassium current, sodium current, GABA_{A} current (inhibitory autapse), and Mcurrent (a slow, noninactivating potassium current), respectively. They take the form
The external forcing term, \(I(t)\), will be described later. Parameter values and units are listed in Table A.1 in Appendix 1. Details of the steady state activation/inactivation functions and time constant functions (\(m_{\infty}\), \(n_{\infty}\), \(h_{\infty}\), \(w_{\infty}\), \(\tau _{n}\), \(\tau_{h}\), \(\tau_{M}\)) are provided in Table A.2 in Appendix 1.
The system (1) features at least two intrinsic timescales. The GABA_{A} autapse has a decay time constant of \(\tau_{d}=9\) ms. Previous works showed that the GABA_{A} synapse is a key ingredient in gamma oscillations [26, 27]. The Mcurrent is proposed in [4] to contribute to a theta rhythm. The time constant of the Mcurrent is voltage dependent and is ten times as long (∼90 ms) as the GABA_{A} decay time near the spiking threshold (around −60 mV).
The cell is receiving external inputs at gamma and theta frequencies. The forcing term \(I(t)\) is
where \(I_{\mathit {ton}}\) is a tonic current, \(I_{\gamma}\) is a purely excitatory pulsatile input which represents the gamma forcing, and \(I_{\theta}\) is a slowly varying sinusoidal current with zero average that represents the theta forcing. \(I_{\mathit {ton}}\) sets the natural spiking frequency of the Icell, that is, the spiking frequency of the Icell without any external theta or gamma forcing. The parameters a and b control the strengths of the gamma and theta forcing. The specific functions and parameters that describe the forcing terms are given in Table A.3 in Appendix 1. At the baseline parameter values, the gamma forcing frequency is 32 Hz and the theta forcing frequency is 4 Hz.
MCurrent Allows the ICell to Follow Gamma Input in the Presence of a Slow Theta Forcing
Simulation results are presented in this section. We will provide some mathematical analysis of these results in Sects. 4 and 5. Simulations of (1), subject to (3), show that, in this particular model, the Mcurrent promotes the oscillator’s entrainment to gamma pulses. In Fig. 1, we consider the response of the Icell when the Mcurrent is present (Fig. 1A) and absent (Fig. 1B). We lowered \(I_{\mathit {ton}}\) in the Icell without Mcurrent such that the natural spiking rate (measured without gamma or theta forcing) is 16 Hz in both cases. Note that, since the theta forcing is slowly changing (in comparison with gamma pulses), we can think of it as a slowly changing tonic current. Then 16 Hz can also be thought of as the natural frequency when the theta forcing is at 0. The most distinctive difference between the two regimes is that the one with Mcurrent is able to follow the pulses but the one without Mcurrent cannot.
In addition to the differences in entrainment, we also find that the voltage envelope of the model with Mcurrent is almost flat, while the voltage envelope of the model without Mcurrent fluctuates with the theta forcing (Fig. 1). This is consistent with the fact that the Mcurrent acts as a homeostatic current with respect to voltage [4]. Here we define homeostasis as the ability of an intrinsic current to counteract subthreshold fluctuations in membrane excitability that come from an external oscillatory forcing.
Interestingly, the homeostatic effect is reduced if the sinusoidal forcing is too fast. Here we simulate (1) with two different values of \(T_{\theta}\) (theta forcing period; see Table A.3). In Fig. 2A, the Icell receives 32 Hz gamma pulses and a 4 Hz sinusoidal forcing (\(T_{\theta }=250\) ms), the same as in Fig. 1A. In Fig. 2B, with a 10 Hz sinusoidal forcing (\(T_{\theta }=100\) ms), the Icell does not follow gamma pulses; the voltage also fluctuates more than with 4 Hz sinusoidal forcing.
However, the homeostasis due to the Mcurrent is not the only reason that the Icell with Mcurrent can follow gamma pulses. Although the natural frequency of the Icells with and without Mcurrent are both 16 Hz (in the absence of theta forcing), the Icell without Mcurrent becomes more excitable at the peak of theta than the Icell with Mcurrent, since it does not have the Mcurrent to oppose the rise of theta forcing. In Fig. 1C, the tonic input is adjusted such that, at the peak of theta, the cell without Mcurrent has the same natural frequency as the cell with Mcurrent (34 Hz, measured without rhythmic inputs but with a constant current at \(I(t)=I_{\mathit {ton}}+b\)). As shown in the blowup in Fig. 1C, during each theta cycle, one spike of the Icell without Mcurrent precedes the forcing. By contrast, the Icell with Mcurrent always spikes after the pulses. This change in spike arrival order has important implications in plasticity [14], since potentiating versus depressing synaptic strengths depends on whether the input spike arrives before or after the target spike (see Discussion). It is important to note that the frequency of gamma input (32 Hz) is slower than the natural frequency of the Icell at the peak of theta forcing (34 Hz). Although it is somewhat intuitive why a faster gamma input could force a slower gamma oscillator, it is not at all obvious how, with Mcurrent, a slower (32 Hz) forcing input can pace a faster oscillator.
We find that the presence of the Mcurrent significantly expands the frequency range where \(1:1\) phaselocking occurs. In Fig. 3, we compare two Icells of natural frequency 34 Hz, with and without Mcurrent, forced by external pulses of frequencies both above and below 34 Hz (but without sinusoidal forcing). The lower bound of the \(1:1\) phaselocking region of the cell without Mcurrent coincides with the cell’s natural frequency (34 Hz). However, the lower bound of the \(1:1\) phaselocking region of the cell with Mcurrent is 29 Hz, slower than the cell’s natural frequency. The upper bound of the \(1:1\) phaselocking region (49 Hz) is the same for both cells, not affected by the Mcurrent. When the external pulses are faster than 49 Hz (last row of Fig. 3), the Icell with Mcurrent can be sparsely entrained: it spikes in alignment with some incoming pulses but skips others. Without the Mcurrent, the Icell develops a slowly increasing phase lag when being forced by pulses faster than the \(1 : 1\) phaselocking range.
In summary, the Mcurrent has two roles in allowing the Icell to follow gamma pulses in the presence of theta forcing. One is the subthreshold homeostatic effect of the Mcurrent, which we will explore in the next section, using averaging theory. The other is the interaction between the Mcurrent and external pulses in the spiking regime. The Mcurrent enables the oscillator to phaselock to a rhythm slower than the oscillator’s natural frequency, and be sparsely entrained by fast inputs. We investigate the reasons for this analytically in Sect. 5 using a reduced model, geometric singular perturbation theory, and bifurcation analysis.
Subthreshold Homeostasis: The Ability of the MCurrent to Stabilize Voltage Is Gradually Reduced as the Sinusoidal Forcing Frequency Increases
Figure 2 shows that the homeostatic effect of the Mcurrent can be sustained with 4 Hz sinusoidal forcing but is reduced with 10 Hz sinusoidal forcing (Fig. 2). To get a quantitative understanding of how the frequency of the forcing influences the homeostasis, we will use averaging theory to average over the output spikes.
It has been shown that the Mcurrent provides the necessary negative feedback effect to create (subthreshold) membrane potential resonance in response to theta inputs [3, 4]. This effect may extend to the spiking regime, but it cannot be captured in a straightforward way using available methods measuring subthreshold resonance. Moreover, the Mcurrent is built up in the spiking regime. Here we use the averaging method [28] to define a new measure to quantify the subthreshold fluctuations of the voltage and gating variables when the neuron is spiking.
To set up the averaged system, we need to identify the fast and slow variables in the system. In (1), the voltage V is the fast variable. All other variables are slow. (Details of the timescale separation can be found in Sect. 5.2.1.) The averaged equations are
where \(x=n,h,s,w\), \(v_{\mathit {rest}}\) is the value of V at the fixed point, \(T(\bar{n}, \bar{h}, \bar{s},\bar{w})\) is the period of the limit cycle, and \(v_{\mathit {spike}}(\bar{n}, \bar{h}, \bar{s},\bar{w},t)\) is the voltage trace during one period of the limit cycle, and the overbar indicates an averaged variable. All functions, expressions of currents, and parameter values are the same as in (1). Note that there are no gamma pulses in (4), because we want to measure the voltage fluctuations in the spiking regime (which are brought about by the sinusoidal current), without the influence of external pulses.
Figure 4 shows that the solution of the averaged system (4) closely approximates the fluctuations of voltage and the Mcurrent gating variable w, under sinusoidal forcing of different frequencies (4 Hz and 10 Hz). Thus, we use the solution of the averaged system (4) as a measure of the overall change of the variables in (1).
The solution of the averaged system (4) reveals a few things that are not obvious from the solution of the original system (1). We observe the following trends as the frequency of the sinusoidal forcing increases (Fig. 5):

The amplitude of the averaged Mcurrent gating variable w̄ decreases.

The averaged voltage is less constant and affected more by the sinusoidal forcing.

The amplitude of the averaged Mcurrent (\(\bar{I}_{M}=g_{M} \bar{w}(E_{M}V)\)) decreases.

The amount (absolute value) of the averaged Mcurrent at the peak of the sinusoidal forcing decreases.

The phase lag (in percentage of the sinusoidal forcing cycle) between the averaged Mcurrent peak and the sinusoidal forcing peak increases.
These results confirms that, in the spiking regime, the Mcurrent cannot keep up with the change in the sinusoidal forcing if the latter is too fast. The homeostatic effect is reduced gradually, as the sinusoidal forcing frequency increases, until it is not enough to keep the cell from spiking before a pulse arrives. The functional consequence of this is that with faster sinusoidal forcing, there is not enough Mcurrent to counteract the amount of excitation at the peak of the sinusoidal forcing.
Spiking Regime: Interaction Between MCurrent and External Pulses
Besides homeostasis, the other function of the Mcurrent that contributes to the entrainment we observed in Fig. 1 (Sect. 3) is the expansion of the frequency range that a cell with Mcurrent can follow. The Mcurrent extends both the lower and the upper limits of the frequency range over which the cell can be entrained. In this section, we explore some mathematical structures that offer insight to the expansion of the entrainment frequency range. First, we use a reduced model to show that having an Mcurrent creates a slow manifold. This model is twodimensional in the subthreshold regime. The reset point is higher from the knee of the Vnullcline when there is an external pulse than when the cell is firing autonomously. Thus, the interspike interval is effectively lengthened, allowing the cell to be phaselocked to a rhythm slower than its natural frequency. Second, utilizing geometric singular perturbation theory, we provide a visualization of the fold structure positions, suggesting that the presence of Mcurrent moves one of the folds such that when the Icell fires, its spike always occur after the external pulse (sparse entrainment). Third, we present a special case where, in a certain parameter regime, the Icell with Mcurrent can phaselock to an arbitrarily slow rhythm due to bistability.
MCurrent Introduces a Slow Manifold and External Pulses Lengthen the Interspike Interval
To capture the essence of the Mcurrent’s role in the system’s dynamics, we analyze a 2D reduced model with artificial spikes. We will show that the Mcurrent introduces a slow manifold and the external pulse changes the reset point position on the slow manifold, enabling the cell with Mcurrent to phaselock to inputs slower than its natural frequency.
The reduced (2D) model approximates the dynamics of the Icell in the subthreshold regime and the onset of spikes. Spikes are added artificially after the voltage leaves the subthreshold regime. This is indicated by a threshold value. The spike width has been set to 1 ms and the spike height varies according to whether the gamma input is present or not to reflect our observations using the full model (see Fig. 1). The reduced model includes the leak current, the Mcurrent and the sodium current where both the activation and the inactivation gating variables are slaved to voltage. The equations describing the subthreshold dynamics (\(V<40\) mV) are as follows.
where
and \(I_{\mathit {app}}\) is a constant. Other functions and parameters are the same as in (1). When V reaches −40 mV, an artificial spike is inserted and the voltage is reset to −64 mV. We make the spike height bigger for a spike triggered by an external pulse than for an autonomous spike as in Fig. 1B. This is essential in reproducing Fig. 1 using the reduced model. We eliminate the autapse to highlight the role of the Mcurrent; we can qualitatively reproduce the results of Figs. 1 and 3 without an autapse. We also eliminate the potassium current and replace \(I_{\mathit {Na}}\) by \(I_{\mathit {NaT}}\) because, in the subthreshold regime, the potassium current is close to zero and the inactivating variable (h) of the transient sodium current is close to its steady state. This change erases the spiking mechanism (except for spike onset) while preserving the subthreshold dynamics.
Without Mcurrent (\(g_{M}=0\)), (5) would be onedimensional (of quadratic integrateandfire type). The derivative of voltage is plotted in Fig. 6A. After a spike has occurred, whether it is autonomous or triggered by an input, the voltage is reset to \(V_{\mathit {reset}}\) (black dot). Then the voltage will keep increasing (since the derivative is positive) until it reaches the threshold and then produces a spike. The time that the voltage takes to rise from \(V_{\mathit {reset}}\) to threshold is the cell’s interspike interval. If the incoming pulses are slower than the natural frequency, i.e. the time between to consecutive pulses is longer than the cell’s interspike interval, the cell will fire before the pulse arrives because the voltage has reached the threshold. Therefore, entrainment is not possible.
Having the Mcurrent in the system introduces a slow manifold (a vicinity of the Vnullcline in Figs. 6B and C). The trajectory travels close to the slow manifold until it reaches the knee. When there is no external pulse (Fig. 6B), the trajectory returns to the reset point (black dot) after a spike. When the spike is triggered by a pulse, however, the reset point is higher up, which is a result of a bigger spike size. If there is no next pulse, it would take the trajectory longer to arrive at the knee in Fig. 6C (triggered spike) than in Fig. 6B (autonomous spike). With Mcurrent, the effective interspike interval when there are external pulses is longer than the natural interspike interval, because of the higher reset point. Hence, the cell with Mcurrent can phaselock to rhythms slower than its natural frequency.
The advantage of using the reduced model is its low dimensionality, which allows us to easily draw and study its phase plane. One limitation of the reduced model is its inability to capture the difference in spike sizes in the presence and absence of external gamma inputs. More specifically, the adjustments are inferred from the simulations of the full model. Thus, the subthreshold dynamics alone do not capture the effect of external pulses on the system. The difference in spike size is crucial; it leads to Mcurrent building up to a higher value during a triggered spike than an autonomous spike. Next, we study a network model that takes external pulses into account to help us understand sparse entrainment.
MCurrent Moves the Fold Structure to Enable Sparse Entrainment
To understand how having the Mcurrent leads to sparse entrainment when the input frequency is greater than the \(1:1\) phaselocking frequency range, we utilize geometric singular perturbation analysis on a network model. First, we formally define the timescales and the associated subsystems, laying the groundwork for our analysis. Then we identify special geometric structures in our model called folds, which correspond to firing thresholds. Next, with some projections, we visualize the fold in a 3D space. We also explain their critical role in phaselocking. Last, the visualizations suggest that Mcurrent alters the position of the Icell firing threshold such that the trajectory of the network always arrives at the Ecell firing threshold before the Icell firing threshold, resulting in entrainment when the input frequency is in an appropriate range and sparse entrainment when the input frequency is above that range.
Geometric Singular Perturbation Analysis
To build a network model, we recast the nonautonomous problem (1) subject to (3) as an autonomous problem by interpreting the gamma pulses as the output of an excitatory cell (Ecell). The Ecell is described by the equations:
where J is a constant current that controls the frequency of the Ecell (external pulses). The leak, potassium, sodium currents are as the same as in (2). The values of maximal conductances and reversal potentials are the same as in (1) and (2), which are provided in Table A.1. At the baseline value \(J=0.5\) μA/cm^{2}, the Ecell is firing at 32 Hz, the same as \(I_{\gamma}\) in (3) in Sect. 2. The Ecell is connected to the Icell through a oneway excitatory synapse. The gating variable of the synapse is defined by
with \(\tau_{r_{e}}=0.1\) ms and \(\tau_{d_{e}}=0.75\) ms. With these modifications, we replace the gamma forcing term \(aI_{\gamma}(t)\) in (3) by
where \(g_{f}=0.5\) mS/cm^{2} and \(E_{e}=0\) mV. Thus, the system we now consider is (1), (7), and (8) subject to (3), where in (3), the gamma forcing term \(a I_{\gamma}(t)\) is replaced by (9) and \(b=0\). We point out that the subsystem (7) describing the Ecell is independent of (1), which is convenient in the analysis, as we will show later.
To uncover the timescales in the network model, we nondimensionalize the system. We choose the reference scales and perform changes of variables as follows: \(k_{v}=100\) mV, \(g_{\mathit {ref}}=10\) mS/cm^{2}, \(k_{t}=\max_{80\le V\le50}(\tau_{M}(V))\approx108\) ms, \(\bar{\tau }_{x}=\max_{80\le V\le50}(\tau_{x}(V))\) for \(x=n,h\), \(\bar{\tau }_{s}=\max(\tau_{r},\tau_{d})=9\) ms, \(\bar{\tau}_{{s_{e}}}=\max (\tau_{r_{e}},\tau_{d_{e}})=0.75\) ms. We let \(V=k_{v}v\), \(V_{2}=k_{v}v_{2}\), \(E_{x}=k_{v}e_{x}\) for \(x=L,K,\mathit {Na},s,M,e\), \(g_{x}=g_{\mathit {ref}}\bar{g}_{x}\) for \(x=n,h,s,w\), \(t=k_{t} \tau\). There is no need to rescale the gating variables (n, h, s, w, \(n_{2}\), \(h_{2}\), \(s_{e}\)), since they are unitless and vary between 0 and 1. The nondimensionalized full (EI) system is
where \(\epsilon=\frac{C}{g_{\mathit {ref}} k_{t}}\approx0.0009 \ll1\), \(\sigma _{1}=\bar{\tau}_{h}/\bar{\tau}_{n}\approx1.4817\), \(\sigma_{2}=\bar {\tau}_{s}/\bar{\tau}_{n}\approx7.7689\), \(\sigma_{3}=\bar{\tau }_{s_{e}}/\bar{\tau}_{n}\approx0.6474\). Note that \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are \(O(1)\). In (10), all state variables and functions on the righthand side are \(O(1)\) with respect to ϵ. The model features two timescales: 1, and \(\frac{1}{\epsilon}\). The two voltage variables v and \(v_{2}\) are fast. All other variables are slow.
Note that the system can be separated into three timescales, with an additional separation between w and other gating variables. However, the third timescale does not bring new information to the geometry, as the superslow manifold is repelling in the relevant parameter range (Appendix 2). Therefore, we use two timescales for our analysis.
With a clear separation of timescales, we utilize geometric singular perturbation theory [29, 30] to study the geometry of the network to understand its dynamics. At the singular limit \(\epsilon\to0\), we attain two subsystems: fast and slow.
The Fast Subsystem
In the fast subsystem, there are two notable structures, the critical manifold and the fold, that have significant functional implications, which we will show in Sect. 5.2.2.
For convenience, let \(x=(v,v_{2})^{T}\), \(y=(n,h,s,n_{2},h_{2},s_{e}, w)^{T}\) denote the fast and slow variables, respectively. The dimensionless system (10) can be concisely written as
where \(F(x,y)=(f_{1}(v,n,h,s,w,s_{e}), f_{2}(v_{2},n_{2},h_{2}))^{T}\), \(G(x,y)=(p_{1}(v,n), \frac{1}{\sigma _{1}} p_{2}(v,h), \frac{1}{\sigma_{2}}p_{3}(v,s), p_{4}(v_{2},n_{2}), \frac{1}{\sigma_{1}}p_{5}(v_{2},h_{2}), \frac {1}{\sigma_{3}}p_{6}(v_{2},s_{e}),g(v,w))^{T}\). With a rescaling of the time variable (\(\tau^{*}=\tau/\epsilon\)) in (11), and then taking the limit \(\epsilon\to0\), we obtain the layer problem with respect to the \(\epsilon\to0\) limit:
where the prime denotes the derivative with respect to the fast time, \(\tau^{*}\). The layer problem describes the fast flow of the singular orbit (the solution at the singular limit). The set of equilibria of (12) is called the critical manifold:
The critical manifold in the \(\epsilon\to0\) limit has attracting and repelling subsets. The attracting and repelling subsets are separated by a submanifold, L, of fold bifurcations of (12) with respect to y, This submanifold L is defined by
Because the Ecell equations in (10) evolve independent of the state variables related to the Icell, the fold condition in (14) simplifies:
That is, the critical manifold \(C_{0}\) has a local turning point whenever either the manifold \(\{f_{1}=0\}\) has a fold with respect to v, or whenever the manifold \(\{f_{2}=0\}\) has a fold with respect to \(v_{2}\). We denote the two branches as Ifold (\(L_{I}\)) and Efold (\(L_{E}\)), respectively:
Also notice that the eigenvalues of \(D_{x}F_{C_{0}}\) are \(\frac {\partial f_{1}}{\partial v} _{C_{0}}\) and \(\frac{\partial f_{2}}{\partial v_{2}} _{C_{0}}\), which are both real. Therefore, there is no Hopf bifurcation in the fast subsystem. Depending on the value of slow variables, the (nonfold) equilibrium of the fast system is a sink, a saddle, or a source.
The Slow Subsystem
Once the singular orbit is on \(C_{0}\), the dynamics is governed by the slow flow, which is described by the reduced problem
Note that the reduced problem can be derived by taking the limit \(\epsilon\to0\) directly from (11).
We also checked if there exist folded singularities (see Definition 8.1.1 in [29]) on L. Folded singularities are generic phenomena in slow/fast systems of the form (10) and give rise to special solutions called canards [29]. Canard solutions stay close to the repelling branch of the slow manifold for some time before moving quickly away. There are models of biological systems where canards are observed [31, 32]. However, in this model, within (and close to) the range of variables and parameters where the singular orbit of the limit cycle typically lies, there are no folded singularities.
Therefore, the fast flow of the limit cycle singular orbit converges to an attracting sheet of \(C_{0}\). Then it follows that the slow flow is governed by the reduced problem (17) until it reaches the fold L. It is well known that the singular slow flow on \(C_{0}\) experiences a finite time blowup at L [29]. Consequently, the limit cycle singular orbit falls off \(C_{0}\) and follows the fast dynamics until it reaches an attracting subset of \(C_{0}\). The fold corresponds to a firing threshold and it is essential in determining if there is phaselocking, which we will further explain in Sect. 5.2.2. We present an example using the 3D Ecell equations to help the reader who is not familiar with this approach to visualize the critical manifold and the fold in Appendix 3.
Folds, Singular Orbits, and Their Implications
In terms of spike order, we can describe sparse entrainment as the Icell spiking (if it does) after the Ecell spikes. Therefore, in a given cycle, the trajectory always crosses the Efold before crossing the Ifold, or the trajectory crosses only the Efold but not the Ifold. We visualize below the Ifold and Efold of two systems with different \(g_{M}\) values to gain insights of how Mcurrent affects the spike order.
Since (10) is ninedimensional, to visualize the folds, we make projections into \((v, v_{2}, s)\) space via approximating the membrane current gating variables as functions of v and \(v_{2}\). We take the following projections based on the separation of timescales:
where \(q(v_{2})\) satisfies \(p_{6}(v_{2}, q(v_{2}))=0\) in (10).
We chose to project these variables because they are the next five fastest variables besides v and \(v_{2}\). These projections will make \(p_{1}(v,n)=p_{2}(v,h)=p_{4}(v_{2},n_{2})=p_{5}(v_{2},h_{2})=p_{6}(v_{2},s_{e})=0\) in (10). After applying (18), the projected fold is now in \((v,v_{2},s,w)\) space. We also fixed w (at some typical value before a spike; see Appendix 5 for details) to visualize a slice of this fourdimensional space in 3D \((v,v_{2},s)\)space, because w is the slowest variable.
We plotted the Ifold and the Efold in the projected space in Fig. 7. Notice that each fold has two sheets, one (at lower voltage value) represents the firing threshold and the other is where the voltage returns after a spike. The twosheet fold structure suggests that the Icell subsystem, although having 5 dimensions, has a similar geometric structure for spiking as the 3D Ecell subsystem, where there are two lines of fold points on the critical manifold, one of which is the firing threshold (shown in Appendix 3).
In Fig. 8A, we plotted the folds of two systems with different \(g_{M}\) values. Their Icell subsystems have the same natural frequency (at \(\epsilon=0.001\)); the two Ecell subsystems are identical. There is sparse entrainment in the high \(g_{M}\) system but not in the low \(g_{M}\) system. Appendix 4 explains how and why we chose these parameter values. We find that increasing \(g_{M}\) moves the Ifold towards the positive v direction. Functionally, the firing threshold of the Icell with high Mcurrent is higher than the one with low Mcurrent, although these two cells have the same excitability (natural frequency).
As the trajectories approach the folds, the low \(g_{M}\) (blue) trajectory hits the (green) Ifold before reaching the Efold: the Icell spikes before the input arrives. By contrast, with high \(g_{M}\), the corresponding (red) Ifold is farther away: the trajectory encounters the Efold first. That is, the Icell with high \(g_{M}\) fires after the input. Moreover, the velocity of high \(g_{M}\) trajectory (red) is slightly slower in the v direction than the velocity of low \(g_{M}\) trajectory (blue), which can be inferred directly from (10). The changes in the Ifold position and trajectory traveling speed result in the change of the first fold encountered by the trajectory.
As a result, with low Mcurrent, the Ifold is closer to the trajectory than in the high Mcurrent case. Therefore, the trajectory may cross the Ifold first, causing the Icell firing before the external pulse arrives. In contrast, the trajectory with high Mcurrent always crosses the Efold first during each input cycle. After that, it may or may not cross the Ifold. Equivalently, in the simulation, we observe sparse entrainment (Sect. 3, last row of Fig. 3A), where the Icell spikes are always in alignment with the input with some missed cycles. During cycles that the Icell does not spike, the Mcurrent gating variable w decreases. The decrease in w causes the Ifold to move towards to the trajectory. However, the amount w decreases during the missed cycle is not more than the amount it increased during the spiking cycles (Fig. 8B). Therefore, when there is sparse entrainment, the Ifold moves further away from the trajectory during consecutive spikes, but shifts back during missed cycles. By the next time the Icell is ready to spike again, the Ifold is still far enough from the trajectory such that the trajectory crosses the Efold first. The sparse entrainment to gamma input in the presence of an Mcurrent is very robust and is observed even for high gamma input frequencies (up to 80 Hz, the maximum input frequency simulated using System (1)).
Bistability of the ICell
We have seen in Sect. 5.1 that Mcurrent helps the oscillator phaselock to a rhythm slower than its natural frequency and in Sect. 5.2 that Mcurrent allows the oscillator to respond to input faster than the \(1:1\) phaselocking range through sparse entrainment. Here, we present a special case where the Icell with Mcurrent is able to \(1:1\) phaselock to an arbitrarily slow rhythm, due to bistability.
MCurrent Allows the ICell with Low Excitation to PhaseLock to an Arbitrarily Slow Input Through Bistability
With low background excitation (\(I_{\mathit {ton}}<5.5\) μA/cm^{2}), the Icell with Mcurrent can phaselock to an arbitrarily slow input. Figure 9 shows a cell with Mcurrent with 16 Hz natural frequency (\(I_{\mathit {ton}}=5\)) being \(1:1\) phaselocked to a 10 Hz input for the first 500 ms. After 500 ms, the forcing is turned off. The cell remains silent instead of returning to its natural 16 Hz oscillation. The internal oscillation of this cell can be turned off by external pulses. Being in the silent state enables the cell to \(1:1\) phaselock to arbitrarily slow external pulses.
Using the numerical continuation software package AUTO [33], we constructed the bifurcation diagram (Fig. 10A) of the Icell with Mcurrent (1) but without gamma or theta forcing, with tonic current as the bifurcation parameter. With the Mcurrent and a low excitation level, a stable limit cycle coexists with a stable fixed point in the bifurcation diagram of the Icell. A family of unstable limit cycles is born from a subcritical Hopf bifurcation (at \(I_{\mathit {ton}}\approx5.6\) μA/cm^{2}). The unstable limit cycle and the stable limit cycle meets at a saddle node of periodics. The Icell ((1) subject to (3) with \(a=b=0\)) with Mcurrent (\(g_{M}=1.5\) mS/cm^{2}) has an interval of bistability between the \(I_{\mathit {ton}}\) values where the saddle node of periodics and the subcritical Hopf bifurcations occur. This is the range of excitation level with which we can observe the shutting off of internal oscillations of the Icell. The bifurcation diagram indicates that the Icell with Mcurrent is a Type II neuron, i.e., the stable limit cycle starts at a nonzero frequency [28, 34].
Without the Mcurrent (\(g_{M}=0\) mS/cm^{2}, Fig. 10B), the Icell does not manifest bistability. The cell’s stable fixed point terminates at a saddle node on invariant circle (SNIC) bifurcation and a stable limit cycle is born. This family of limit cycles starts at infinite period, i.e., zero frequency. Hence, the cell without Mcurrent is a Type I neuron [28, 34]. The left bifurcation diagram transitions smoothly into the right as \(g_{M}\) decreases. The Hopf bifurcation point becomes a SNIC through a Bogdanov–Takens bifurcation.
Therefore, with baseline parameters, bistability exists when there is an Mcurrent and the excitation is within the range where a fixed point and a stable limit cycle coexist. These are the conditions of the Icell being entrained by arbitrarily slow inputs.
Possible Ways to Expand the Bistability Range
In Fig. 10A, the bistability region under default parameters spans over about 0.86 μA/cm^{2}, which is quite small. We explore how changes in the parameters can affect the size of this bistability window. In AUTO, we continue the Hopf bifurcation point and the starting point of the stable limit cycle in parameters \(g_{L}\), \(g_{K}\), \(g_{\mathit {Na}}\), \(g_{s}\), and \(g_{M}\).
We show below that a reasonable increase in the maximal conductance of the Mcurrent or the potassium current will expand the bistability window, while other parameters either do not expand the window much, or need to be increased dramatically to achieve the effect. Figures 11A–E show how the Hopf bifurcation point and the stable limit cycle starting point move as we move away from the baseline values for the maximal conductances (\(g_{L}\), \(g_{K}\), \(g_{\mathit {Na}}\), \(g_{s}\), and \(g_{M}\)). The two points move at different rates. It is not clear when the size of the bistability window is at its maximum from Figs. 11A–E. Therefore, we compute the vertical distance between the red curve and the blue curve for each subplot and put them together in one graph (Fig. 11F). Changes in \(g_{K}\), \(g_{s}\), and \(g_{M}\) increase the bistability window size significantly. However, \(g_{s}\) needs to be at least 10 times of its baseline value to make the bistability window size larger than 2.5 μA/cm^{2}, which is the value near the plateau of the \(g_{s}\) curve in Fig. 11F. When \(g_{M}\approx 3.2066\) mS/cm^{2}, a little more than doubling the baseline value, the bistability window size is at its maximum (around 4.4303 μA/cm^{2}). The size of the bistability window is also larger than 3 μA/cm^{2} when 20.1 mS/cm^{2}\(< g_{K}<25.8\) mS/cm^{2}, less than tripling the baseline value (\(g_{K}=9\) mS/cm^{2}). Therefore, increasing the maximal conductance of the Mcurrent or the potassium current can most easily make the Icell be bistable subject to a larger range of background excitation.
Bistability is of functional significance as it permits an otherwise oscillating neuron to be entrained by an arbitrarily slow spike train. The phaselocking range is dramatically expanded. The phaselocking is also more robust: if the forcing frequency changes abruptly (but within the \(1:1\) phaselocking range), the cell can follow without delay.
hCurrent Also Allows the Cell to PhaseLock to Rhythms Slower than Its Natural Frequency, but Spikes Occur Before Inputs
In models of hippocampal gamma and theta oscillations, the intrinsic theta timescale comes from an intrinsic membrane current, often an hcurrent or an Mcurrent [3, 4, 18, 35, 36]. The Mcurrent and the hcurrent are functionally similar in that they both oppose changes in voltage in response to an applied current at a slow timescale. The hcurrent is an inward current that activates with hyperpolarization, whereas the Mcurrent is an outward current that decreases with hyperpolarization.
We found that replacing the Mcurrent by the hcurrent (Appendix 6) can reproduce some but not all results. Like the Mcurrent, the hcurrent allows the cell to \(1:1\) phaselock to inputs slower than the cell’s natural frequency. Figure 12A shows a cell with hcurrent with 34.5 Hz natural frequency can \(1:1\) phaselock to 32 Hz input. In fact, it is able to \(1:1\) phaselock to inputs as slow as 30 Hz. However, all spikes in Fig. 12A occur consistently before the input, whereas with Mcurrent, the spikes always occur after the input. As a result, when the theta input (sinusoidal current) is added, spikes that occur near the peak of theta input precede the external pulses, yet the spiking frequency of the cell with hcurrent is the same as the external pulses (Fig. 12B). We were not able to obtain spikes occurring after inputs using the hcurrent for a wide range of strengths of the hcurrent simulated. It is unclear what mathematical aspect is responsible for the differences we observe in the systems with Mcurrent and with hcurrent. It could be because the disparity between the Mcurrent being activated after a spike and the hcurrent being deactivated after a spike. More work needs to be done to understand the difference and its functional implications.
Discussion
In this paper, we consider a small model network consisting of a single cell with an inhibitory synapse. The network has both gamma (inhibitory synapse) and theta (Mcurrent) timescales, and is forced by a combination of spiking gamma input and sinusoidal theta input. The Mcurrent allows the network to \(1:1\) phaselock to a wider range of frequencies than a network without the Mcurrent but with matched natural frequency. It also provides homeostasis against the oscillating theta input, making the frequency of the network less variable over a theta cycle. However, this homeostatic effect works only when the sinusoidal forcing is of theta or slower frequencies. Additionally, when responding to pulsatile gamma inputs too fast to follow cycle by cycle, the system with the Mcurrent exhibits sparse entrainment. By contrast, the system without the Mcurrent (but with the same natural frequency) reacts to faster gamma inputs by phase walkthrough.
The reduced model with artificial spikes reveals that the external pulses push the reset point of the limit cycle to a higher place on the slow manifold than when there is no external pulse, effectively extending the interspike interval. To understand sparse entrainment, which requires studying both inputs and outputs, we utilized geometric singular perturbation theory on the EI network model. We showed that the timing of the singular orbit crossing the branches of the fold is essential to determining whether there is entrainment in the system at its singular limit. The position of the fold is influenced by the maximal conductance of the Mcurrent and therefore alters the spike order. The Mcurrent also changes the bifurcation structure such that the system becomes bistable, which further increases the robustness of the system’s ability to be entrained to external pulses. Bifurcation analysis and numerical continuation show that increasing the maximal conductance of the Mcurrent or the potassium current expands the bistability region significantly.
In the present work, we explored how internal network theta and gamma timescales interact with external theta and gamma forcing. Interestingly, we found that it is the intrinsic network theta timescale that allows the intrinsic gamma frequency spiking to coordinate precisely with the excitatory gamma forcing input. Note that although the internal theta timescale is also the subthreshold resonance frequency of the cell (due to the Mcurrent), this does not automatically imply suprathreshold resonance or precise response spiking [38]. The phenomenon we are studying in this paper is the phaselocking of spikes, not subthreshold resonance. Precisely timed spiking is critical to spiketimedependent plasticity, a mechanism thought to be important to learning and memory [14].
To the best of our knowledge, this is the first demonstration of the phenomenon that the Mcurrent, which has a theta timescale, contributes to the precise coordination of spikes of gamma frequency. Since the Mcurrent is a noninactivating potassium current with a slow decay time constant, it is active in the subthreshold range of membrane voltages and is able to influence the membrane potential of neurons during the interspike interval [28, 39]. Our analysis of how the Mcurrent allows the cell to be entrained by rhythms slower than the cell’s natural frequency indicates that the Mcurrent works to prolong the duration of the intrinsic network interspike interval when excitatory gamma forcing is present. Prolongation of the intrinsic network interspike interval allows excitatory gamma input to force the network to entrain to a lower gamma frequency than its intrinsic gamma oscillation frequency.
The prolongation of the intrinsic network interspike interval by gamma forcing is observed from two perspectives. One is the higher reset point after each spike triggered by external pulses indicated by the reduced model. The other perspective is from the timing of spike onset inferred from the geometry of the network system. We saw a relative shift in the positions of the Icell fold structure with respect to the singular orbit of the limit cycle. The Mcurrent shifts the Icell fold further away than the Efold from the singular orbit, thus allowing the forcing input (Ecell input) to pace the system. Here we do not consider folded singularities on the fold structures because, although folded singularities can constrain dynamics near them; in our case there were no such singularities near the relevant physiological parameters.
We also observed that the homeostatic effect of Mcurrent is gradually reduced as the frequency of the sinusoidal current increases in Sect. 4. The fastest sinusoidal current that can be imposed onto the cell while preserving the cell’s entrainment to gamma pulses seems to be related to the subthreshold resonance frequency of the cell as a result of having the Mcurrent. However, we currently do not have a way to mathematically explain the connection between the two, since it is not clear how the subthreshold resonant properties of individual cells are communicated to the spiking regime in the presence of a limit cycle. Further research is needed to address this question, not only for the specific case studied here, but also in a general context.
Additionally, we find that the Mcurrent provides a region of bistability to the network system such that the network can phaselock to an arbitrarily slow forcing input by turning off its internal oscillation. Previous works have shown that the Mcurrent alters the neuron’s firing mechanism. In [40], Ermentrout et al. reported that, without the Mcurrent, the neuron’s stable limit cycle is created at a saddlenode point. With the Mcurrent, however, the stable fixed point loses its stability at a Hopf bifurcation. Thus, the addition of the Mcurrent changes the neuron from Type I to Type II. Acker et al. [37] also reported on the switch of neuron type due to the addition of a slow resonating current (an hcurrent or a slow, noninactivating potassium current, e.g., Mcurrent). Moreover, in Sect. 4.4.2 of [41], Ermentrout and Terman also addressed how adding the Mcurrent allows a SNIC to transition to Hopf through a Bogdanov–Takens bifurcation. However, these results do not discuss whether there is bistability, or the implication of bistability on the system (allowing the cell to be entrained by an arbitrarily slow rhythm), as we have shown here.
Last, we addressed what happens when we replace the Mcurrent by the hcurrent. While both Mcurrent and hcurrent are able to allow the cell to be entrained by a rhythm slower than its natural frequency, the resulting spike orders are different. With an Mcurrent, the Icell follows the input, while the Icell precedes the input with an hcurrent. It suggests that the Mcurrent and the hcurrent have very different functional implications for plasticity. In the case of the Mcurrent, the cell spiking after the input would strengthen the synaptic connection; whereas with an hcurrent, the synaptic connection would be depressed due to the cell spiking before the input [14]. We were able to reproduce Fig. 12 using a reduced model with hcurrent (similar to the one used in Sect. 5.1), but the phase plane of the reduced model with hcurrent does not offer obvious explanations of the spike order difference observed in the models with Mcurrent and with hcurrent.
The major finding of this work is that the theta timescale properties of the Mcurrent allow this EI network to precisely coordinate with external forcing on a gamma timescale. This work shows how the interaction of currents on different timescales can be of functional importance in network level phenomena.
Abbreviations
 CFC:

crossfrequency coupling
 Ecell:

excitatory cell
 GABA:

gammaaminobutyric acid
 Icell:

oscillator with an inhibitory autapse
 PING:

pyramidalinterneuronal network gamma
 SNIC:

saddle node on invariant circle
 STDP:

spiketimedependent plasticity
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Acknowledgements
The authors thank Christoph Börgers for providing the code of the mathematical model.
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This work is supported by National Science Foundation (NSF) grants DMS1225647 and DMSDMS1514796.
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NK and MM designed the project. YZ carried out the analysis with help from TV and HR. All authors wrote the paper. All authors read and approved the final manuscript.
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Appendices
Appendix 1: Parameters, Units, and Functions in the Mathematical Model
Here we provide the parameter values and expressions of auxiliary functions used in the mathematical model. In particular, Table A.1 contains baseline parameter values for (1) and (7). Expressions for infinity and τ functions in Eq. (1) can be found in Table A.2. Functions and parameters that describes the forcing terms are given in Table A.3.
Appendix 2: ThreeTimescale Separations
We showed in Sect. 5.2.1 that (1) has two timescales through nondimensionalization. However, the system can be separated further into three timescales as follows.
where \(\epsilon_{2}=\frac{C}{g_{\mathit {ref}} \bar{\tau}_{n}}\approx0.0863 \ll1\), \(\delta=\bar{\tau}_{n}/k_{t}\approx0.0106 \ll1\), and all other variables are defined as in Sect. 5.2.1. Here, all state variables and functions on the righthand side are \(O(1)\) with respect to \(\epsilon_{2}\) and δ. System (19) features a cascade of timescales: 1, \(\frac{1}{\delta }\), and \(\frac{1}{\epsilon_{2}\delta}\). The two voltage variables v and \(v_{2}\) are fast. All gating variables except w are slow. The Mcurrent gating variable w is superslow.
With three timescales, there two ways to take the singular limit: \(\epsilon_{2}\to0\) and \(\delta\to0\). The fast subsystem is the same as in Sect. 5.2.1 by taking \(\epsilon_{2}\to0\). As the singular orbit lands on \(C_{0}\), the critical manifold, it switches to the slow flow, which itself is a slow/fast system with timescales separated by δ. Define the superslow manifold as
where \(G_{1}\) is a vector of the first six elements of G (see Sect. 5.2.1). Note that \(C_{0}^{ss}\) is onedimensional and repelling in the parameter range we study. This timescale separation by δ does not bring new geometric insight about the slow flow on \(C_{0}\). Thus, we focus only on the singular limit \(\epsilon_{2}\to0\), which is the same as the singular limit \(\epsilon\to0\) in (10).
Appendix 3: Example: ECell
Here we use the Ecell subsystem to provide an example of visualizing the geometry of the system at its singular limit. The low dimensionality of the Ecell system allows us to easily visualize how the fold acts as a firing threshold. The dimensionless equations of the Ecell are
In the \(\epsilon\to0\) limit, the critical manifold of the Ecell is
The fold on \(C_{0}^{E}\) with respect to the slow variables \(n_{2}\), \(h_{2}\) is
We use superscript \(L^{E}\) to distinguish the fold of the Ecell subsystem (20) from the Efold branch \(L_{E}\) of the network model (10).
In Fig. 13, we show the Ecell critical manifold \(C_{0}^{E}\) (blue surface), Ecell fold \(L^{E}\) (red lines), and the singular orbit (black curve) from two perspectives. The slow part of the singular orbit moves on \(C_{0}^{E}\) and the orbit jumps at \(L^{E}\) to the opposite attracting sheet of \(C_{0}^{E}\). Therefore, one branch of the fold \(L^{E}\) (the one that the singular orbit crosses as \(v_{2}\) increases) is equivalent to a physiological firing threshold. This allows us to identify a geometric structure, \(L^{E}\), which can be used as the separation between spiking and nonspiking behavior of the neuron.
The true singular orbit of the limit cycle can be well approximated by a numerically simulated one. In Fig. 13B, the numerically approximated singular orbit is obtained by simulating (20) with \(\epsilon=0.001\). Note that this approximated orbit stays very close to the real singular orbit of the limit cycle, which can be constructed by concatenating the slow flow and the fast flow. We used the approximated singular orbit instead of the real singular orbit in Fig. 8. We do this because for fair comparison, we need to adjust \(I_{\mathit {ton}}\) to ensure the Icells with and without Mcurrent have the same natural frequency. However, at the limit \(\epsilon\to0\), frequency loses its physical meaning. The \(\epsilon=0.001\) is used only for approximating singular orbits. All other geometric structures (i.e. critical manifold, folds) are defined at the \(\epsilon\to0\) limit.
Appendix 4: Choice of \(g_{M}\) and J in Fig. 8
Figure 8 provides a visualization of the multidimensional geometric structure of the fold manifolds of (10), which is essential in understanding how the Mcurrent affects the phaselocking. Since the fold manifolds are sixdimensional, it requires careful choices of approximations and projections to present them in a space of lower dimension (the threedimensional \((v,v_{2},s)\) space). In this section, we list the values of \(g_{M}\) and J used to generate Fig. 8 and explain why we choose them, for the purpose of reproducing the results.
Here we define a particular value of \(g_{M}\) as the boundary between phaselocking and not phaselocking. We have seen in Fig. 1 that the Icell with Mcurrent (\(g_{M}=1.5\) mS/cm^{2}) can phaselock to gamma pulses, while the Icell without Mcurrent (\(g_{M}=0\) mS/cm^{2}) cannot. There exists a value \(g_{M}^{*}\) such that the Icell can phaselock to the external pulses if and only if \(g_{M}> g_{M}^{*}\). In the EI model (10), we find, numerically, that \(g_{M}^{*}\approx0.67\) mS/cm^{2}. This was done by first finding the stable limit cycle of (10) with original parameters. Next, in AUTO, we continue this stable limit cycle in parameter \(g_{M}\) to obtain the bifurcation diagram in Fig. 14. The stable limit cycle at \(g_{M}=1.5\) mS/cm^{2} continues as \(g_{M}\) decreases and terminates at \(g_{M}^{*}\approx 0.67\) mS/cm^{2}.
Notice that there is another branch of stable limit cycles that is not connected with the curve of stable limit cycles continued from \(g_{M}=1.5\) mS/cm^{2}. Although this set of stable limit cycles also represents \(1:1\) phaselocking between the Icell and the Ecell, the Icell consistently spikes before the Ecell instead of right after the Ecell (Fig. 14B). We call this the “phase advance \(1:1\) phaselocking.” This regime occurs in a much smaller parameter window compared with the other regime, which we saw in Fig. 1A (call it the “phase delay \(1:1\) phaselocking”). Moreover, both phase advance and phase delay \(1:1\) phaselocking stable limit cycles exist when 0.67 mS/cm^{2}\(< g_{M}<0.72\) mS/cm^{2}. It depends on the initial condition to which regime the system will converge. Since the phase delay \(1:1\) phaselocking regime is more typical, we consider the value where the phase delay \(1:1\) phaselocking breaks as the boundary between “Icell follows Ecell” and “Icell cannot follow Ecell.” Since we consider the \(\epsilon\to0\) limit in Sect. 5.2.1, we continue the boundary \(g_{M}^{*}\) in two parameters \(g_{M}\) and ϵ in AUTO. As \(\epsilon\to0\), the boundary \(g_{M}^{*}\) shifts to negative values (Fig. 15A). Although it is out of the physiologically reasonable range, it does not affect the mathematical analysis.
The network model (10) and its singular limit do not have the same boundary value \(g_{M}^{*}\). The parameter J of the Ecell also changes in the singular limit. We know from Fig. 14A that the \(1:1\) phaselocking range of (10) starts at \(g_{M}^{*}\approx 0.67\) mS/cm^{2} and \(J=0.5\) μA/cm^{2}. Figure 15A shows that \(g_{M}^{*}\) shifts to negative values when \(\epsilon\to0\). For a given input (fixed J), there exists a boundary value \(g_{M}^{*}\) where \(1:1\) phaselocking starts. Similarly, given a fixed \(g_{M}\), there exists a boundary value for J (controlling the input frequency) where \(1:1\) phaselocking starts. As approaching the singular limit \(\epsilon\to0\), both boundary values change. Therefore, we also continued \(g_{M}^{*}\) in parameters ϵ and J, to obtain the J value at the boundary at the singular limit. Figure 15B indicates that at the singular limit \(\epsilon\to0\), the boundary value for J is slightly less than 0.2. Note that the parameters mentioned from now on do not have units, because we are considering the singular limit of a dimensionless system.
To approximate the singular orbit of (10) in Fig. 8, we use \(\epsilon=0.001\) in the simulations. According to Fig. 15, we need another set of \(g_{M}\), J, and \(I_{\mathit {ton}}\) values to simulate the phaselocking and nonphaselocking scenarios. In Fig. 8B, we control the input frequency by fixing \(J=0.16\). The two levels of \(g_{M}\) we chose are −0.3 and −0.5. In order to ensure that the natural frequencies of the Icell are the same at these two levels of \(g_{M}\) and \(\epsilon=0.001\), we set \(I_{\mathit {ton}}=1\) when \(g_{M}=0.5\) and \(I_{\mathit {ton}}=5\) when \(g_{M}=0.3\). Using these parameters, the Icell phaselocks to the Ecell (with \(J=0.16\)) at the high Mcurrent level (\(g_{M}=0.3\), \(I_{\mathit {ton}}=5\)), but does not phaselock to the Ecell at the low Mcurrent level (\(g_{M}=0.5\), \(I_{\mathit {ton}}=1\)).
Appendix 5: Choice of Fixed w in Fig. 8
To project the folds (16) in \((v,v_{2},s)\) space, we need to fix a w value in addition to the approximations we made in (18). Since we want to know what contributes to the timing of the spiking of the Icell, we chose a typical w value before the Icell spikes. Figure 16 shows two simulations of the approximated singular orbit (\(\epsilon=0.001\)) with two levels of maximal Mcurrent conductance. With low \(g_{M}\) (Fig. 16A), w is around 0.74 before spiking. With high \(g_{M}\) (Fig. 16B), w is around 0.7 before spiking. Hence, in Fig. 8B, we fix \(w=0.74\) for \(g_{M}=0.5\) (green Ifold) and \(w=0.7\) for \(g_{M}=0.3\) (red Ifold).
Appendix 6: Equations of the hCurrent
The equations of the hcurrent we use in the simulations are as follows. They are the same as in [37].
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Zhou, Y., Vo, T., Rotstein, H.G. et al. MCurrent Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains. J. Math. Neurosc. 8, 13 (2018). https://doi.org/10.1186/s1340801800686
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DOI: https://doi.org/10.1186/s1340801800686
Keywords
 Phaseamplitude coupling
 Theta rhythm
 Geometric singular perturbation theory
 Averaging
 Bistability
 Multiple timescales
 Biophysical modeling