# Modeling Excitable Cells with Memristors

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## Abstract

**:**

## 1. Introduction

^{+}and Ca

^{2+}to enter the cell and voltage-sensitive K

^{+}channels and voltage-insensitive K

^{+}channel that allow K

^{+}ions to leave the cell and activate intracellular calcium ions, respectively [27,28,29]. Therefore, the outward current carried by K

^{+}ions pass through the voltage and calcium-sensitive channels, and the inward currents carried by Na

^{+}and Ca

^{2+}ions pass through the voltage-sensitive Na

^{+}and Ca

^{2+}channels. However, the above models consist of complicated nonlinear differential equations associated with membrane voltage. Later, a modified model was presented by Chay [30], assuming the β-cells of the voltage-sensitive Na

^{+}conductance are almost inactive and the inward current is almost exclusively carried by Ca

^{2+}ions through the voltage-sensitive Ca

^{2+}channel. Therefore, the assumption of a mixed effective conductance was formulated without affecting the results by expressing the total inward current in terms of a single mixed conductance g

_{I}and the reversal potential E

_{I}of the two functionally independent Na

^{+}and Ca

^{2+}channels. The model consists of three nonlinear differential equations, in contrast to the other complicated models of an excitable membrane of a pancreatic β-cell. Our studies in this paper typically focus on the simplified Chay neuron model of an excitable cell [30].

_{I}functions as a nonlinear resistor, while the state-dependent voltage-sensitive potassium ion-channel g

_{K,V}and calcium-sensitive potassium ion-channel g

_{K,Ca}function as time-invariant memristors in Chay neuron model of an excitable cell. Additionally, this study seeks to delve into the memristive model of an excitable cell with the goal of gaining deeper insights into the dynamic behavior of excitable cell through analyses of small signal equivalent circuit models, pole-zero diagrams, local activity principle, edge of chaos and Hopf bifurcation theorem.

## 2. Chay Neuron Model of an Excitable Cell

^{+}), potassium (K

^{+}), calcium (Ca

^{2+}), and chloride (Cl

^{−}) across the membrane, leading to changes in the cell’s membrane potential and the generation of electrical signals. The study of excitable cells encompasses a wide array of topics, and our primary aim is to present a unified model for both neuronal and secretory excitable membranes based on the Chay neuron model. The Chay neuron model, which focuses on a simplified representation of neuronal and secretory excitable membranes, aims to provide a unified framework for understanding the complex electrical activity observed in excitable cells. This model typically involves just three ordinary differential equations (ODEs) to capture the essential features of an excitable cell membrane. The model consists of (a) mixed ion channel g

_{I}; (b) state-dependent voltage-sensitive potassium ion channel g

_{K}

_{,V}; (c) calcium-sensitive potassium ion channel g

_{K,Ca}; and (d) leakage channels, which are described by the following differential equations:

_{m}, potentials E

_{I}, E

_{K}, and E

_{L}for mixed Na

^{+}-Ca

^{2+}ions, K

^{+}, and leakage ions, respectively. The conductances g

_{I}, g

_{K}

_{,V}, g

_{K}

_{,Ca}, and g

_{L}represent the voltage-sensitive mixed ion channel, voltage-sensitive potassium ion channel, calcium-sensitive potassium ion channel, and leakage channel, respectively. In the upcoming session, we will provide rigorous proofs that the state-independent mixed ion channel functions as a nonlinear resistor. However, the commonly held belief regarding state-dependent ion channels exhibiting time-varying conductances are found to be conceptually incorrect from the perspective of electrical circuit theory. Contrary to this conventional assumption, these ion channels do not adhere to time-varying conductance principles. Instead, they align more accurately with the characteristics of time-invariant generic memristors from a circuit-theoretic standpoint. A rigorous proof will be demonstrated in the subsequent section. The parameters for this model are summarized in Table 1 (The units of conductance for mixed ion channel, voltage-sensitive potassium ion channel, calcium-sensitive potassium ion channel and leakage ion channel in the Chay model [30] are assumed as $g*=\frac{conductance}{membranecapacitance}=\frac{mS/c{m}^{2}}{mF/c{m}^{2}}=\frac{1}{second(s)}={s}^{-1}$. As, we are assuming the value of membrane capacitance (C

_{m}) = 1 mF/cm

^{2}, we use the unit of all the conductances of the ion channels, g = mS/cm

^{2}, throughout this study, which is also the equivalent unit g* of the original Chay model) and a list of abbreviations of the model parameters are illustrated in Appendix A. The comparison analyses of the HH model [1], FitzHugh–Nagumo model [31], ML model [8], and Chay model [30] are summarized in Table 2 along with their respective strengths and limitations. It is notable that each model possesses distinct advantages and drawbacks, making them suitable for different research contexts and questions. The choice of the model depends on the level of detail required, the computational resources available, and the specific phenomena under investigation. This study predominantly centers on the Chay neuron model of excitable cells.

## 3. Pinched Hysteresis Fingerprints of the Ion Channel Memristor

#### 3.1. Voltage-Sensitive Mixed Ion channel Nonlinear Resistor

_{I}and current i

_{I}in the second element (from left) in Figure 1a is given by

_{I}(m

_{∞}, h

_{∞}), where m

_{∞}and h

_{∞}are the functions of the voltage v

_{I}across the two-terminal element.), as depicted in the second element (from the left) in Figure 1b. To verify the voltage-sensitive mixed ion channel is a nonlinear resistor, an extensive numerical simulation for a sinusoidal input voltage source v

_{I}= 100sin(2πft) mV is performed for the three different frequencies, namely, f = 100 Hz, 200 Hz, and 1 KHz, respectively. Figure 2 shows the corresponding output nonlinear waveform on the current i

_{I}vs. voltage v

_{I}plane for these frequencies, confirming that the mixed ion channel exhibits the properties of a nonlinear resistor.

#### 3.2. Voltage-Sensitive Potassium Ion Channel Memristor

_{K,V}, and the current is i

_{K,V}. Then

_{K}

_{,V}(t) = 100sin(2πft) mV is applied to the voltage-sensitive potassium ion channel with frequencies f = 100 KHz, 500 KHz, and 4 MHz, respectively. As shown in Figure 3, the zero-crossing pinched hysteresis loops shrink as the frequencies increase and tend to a straight line at 4 MHz, which confirms that the voltage-sensitive potassium ion channel is a generic memristor. All of these pinched hysteresis loops exhibit the fingerprints of a memristor [33].

#### 3.3. Calcium-Sensitive Potassium Ion Channel Memristor

_{K}

_{,Ca}(Since the same potential E

_{K}is shared by the voltage-sensitive potassium ion channel memristor and calcium-sensitive potassium ion channel memristor, the voltage assumed, V − E

_{K}= v

_{K,V}in Equation (16) and V − E

_{K}= v

_{K,Ca}in Equation (21), are identical. The voltages v

_{K,V}and v

_{K,Ca}are assumed to distinguish the input voltage applied to the voltage-sensitive potassium ion channel memristor and the calcium-sensitive potassium ion channel memristor, respectively.) and the current is i

_{K}

_{,Ca}. The current entering the channel is given by

_{KCa}(t) = 100sin(2πft) mV with frequencies f = 10 Hz, 30 Hz, and 200 Hz, respectively. Observe from Figure 4 that all the zero-crossing pinched hysteresis loops shrink as the frequencies of the input signal increases and tend to a straight line for the frequency f = 200 Hz. All of the pinched hysteresis fingerprints confirm that the calcium-sensitive potassium ion channel is a generic memristor.

## 4. DC Analysis of the Memristive Chay Model of an Excitable Cell

_{m}, gate activation n of the voltage-sensitive potassium ion channel memristor, and concentration of calcium-sensitive Ca of the calcium-sensitive potassium ion channel memristor to zero from Equations (1), (2), and (3), respectively. By determining these equilibrium points, insights can be gained into the behavior of the excitable cell under different conditions, such as varying input stimuli or parameter values, and can be expressed as a function of current I as follows:

_{m}in Equation (26) gives the explicit formula of the DC V-I curve of the memristive Chay model. We have plotted the individual DC V-I curves of the voltage-sensitive mixed ion channel non-linear resistor, voltage-sensitive potassium ion channel memristor, calcium-sensitive potassium ion channel memristor, and leakage channel at equilibrium voltages V

_{I}, V

_{K}

_{,V}, V

_{K}

_{,Ca}and V

_{L}, as shown in Figure 5b, Figure 5c, Figure 5d, and Figure 5e, respectively. Figure 5f shows the DC V-I curve of Figure 5a over the range of DC voltage −50 mV < V

_{m}< −24 mV. For any DC value of V

_{m}, we calculated the corresponding value of I as the vertical axis. Our extensive calculations show that the two Hopf bifurcation points occur at V

_{m}= −48.763 mV (resp., I = −66.671 µA) and V

_{m}= −27.984 mV (resp., I = 433.594 µA), respectively. Details of these two bifurcation points will be discussed in upcoming section.

## 5. Small-Signal Circuit Model

#### 5.1. Small-Signal Circuit Model of the Mixed Ion Channel Nonlinear Resistor

_{I}(The equilibrium point Q

_{I}at v

_{i}= V

_{I}is obtained by solving Equation (12)) on the DC V

_{I}-I

_{I}curve is derived as follows:

_{I}, we get

_{I}(s; Q

_{I}) of the small-signal equivalent circuit of the voltage-sensitive mixed ion channel nonlinear resistor at the DC operating point Q

_{I}is given by

_{12}(Q

_{I}) and resistance R

_{1,I}as a function of the DC equilibrium voltage V

_{I}= V

_{m}− E

_{I}, where E

_{I}= 100 mV, are shown in Figure 6a and Figure 6b, respectively. The explicit formulas for computing coefficient a

_{12}(Q

_{I}) are given in Figure 7 for readers’ convenience.

#### 5.2. Small-Signal Circuit Model of the Voltage-Sensitive Potassium Ion Channel Memristor

_{K}

_{,V}(The equilibrium point Q

_{K,V}at v

_{K,V}= V

_{K,V}is obtained from Equation (19) by solving f(n; V

_{K,V}) = 0 for n = n

_{K,V}. The explicit formula for n(V

_{K,V}) is given in Figure 11) on the DC V

_{K}

_{,V}-I

_{K}

_{,V}curve is derived by defining

_{K}

_{,V}), V

_{K}

_{,V}(Q

_{K}

_{,V})), we obtain

_{K}

_{,V}), V

_{K}

_{,V}(Q

_{K}

_{,V})), we obtain

_{K}

_{,V}(s; Q

_{K}

_{,V}) for the small-signal equivalent circuit of the voltage-sensitive potassium ion channel memristor at equilibrium point Q

_{K}

_{,V}:

_{11}, a

_{12}, b

_{11}, b

_{12}, inductance L

_{K}

_{,V}, resistance R

_{1K,V}, and resistance R

_{2K,V}as a function of the DC equilibrium voltage V

_{K,V}= V

_{m}− E

_{K}, where E

_{K}= −75 mV are shown in Figure 9 and Figure 10, respectively. Please note that the local activity, edge of chaos 1, and edge of chaos 2, shown in Figure 10a–c, are not the local activity and edge of chaos domains of the separate two terminals of the voltage-sensitive potassium ion channel memristor. The small signal positive inductance and resistances (i.e., L

_{K}

_{,V}> 0, R

_{1K,V}> 0, and R

_{2K,V}> 0) of the potassium ion channel memristor observed over the local activity, edge of chaos 1 and edge of chaos 2 regime are just the corresponding range of the voltage with respect to V

_{K}

_{,V}of the entire connected Chay small-signal equivalent circuit of Figure 1b and Figure 16. For the readers’ convenience, the explicit formulas for computing the coefficients a

_{11}(Q

_{K}

_{,V}), a

_{12}(Q

_{K}

_{,V}), b

_{11}(Q

_{K}

_{,V}), b

_{12}(Q

_{K}

_{,V}), and L

_{K}

_{,V}, R

_{1K,V}, and R

_{2K,V}are summarized in Figure 11.

#### 5.3. Small-Signal Circuit Model of the Calcium-Sensitive Potassium Ion Channel Memristor

_{K}

_{,Ca}(The equilibrium point Q

_{K,Ca}at v

_{K,Ca}= V

_{K,Ca}is obtained from Equation (23) by solving f(Ca; V

_{K,Ca}) = 0 for Ca = Ca

_{K,Ca}. The explicit formula for Ca(V

_{K,Ca}) is given in Figure 15.) in the DC V

_{K}

_{,Ca}-I

_{K}

_{,Ca}curve is derived by defining

_{K.Ca}), V

_{Ca}(Q

_{K}

_{,Ca})), we obtain

_{,K,Ca}), V

_{Ca}(Q

_{,K,ca})), we obtain

_{K}

_{,Ca}(s; Q

_{K}

_{,Ca}) of the small-signal equivalent circuit of the calcium-sensitive potassium ion channel memristor at equilibrium point Q

_{K}

_{,Ca}:

_{11}, a

_{12}, b

_{11}, b

_{12}, and inductance L

_{K}

_{,Ca}, resistance R

_{1K,Ca}, and resistance R

_{2K,Ca}as a function of the DC equilibrium voltage V

_{K}

_{,Ca}are shown in Figure 13 and Figure 14

**,**respectively. The small-signal inductance and resistances (i.e., L

_{K}

_{,Ca}> 0, R

_{1K,Ca}> 0 and R

_{2KCa}> 0) over the edge of chaos 1 and edge of chaos 2 with respect to the V

_{K}

_{,Ca}are shown in Figure 14a, Figure 14b, and Figure 14c, respectively. Please note that the local activity, edge of chaos 1 and edge of chaos 2, shown in Figure 14a–c, are not the local activity, edge of chaos 1 and edge of chaos 2 of the individual calcium-sensitive potassium ion channel memristor. The local activity and edge of chaos domains are just information showing the corresponding range of voltage with respect to V

_{K}

_{,Ca}when measured across the individual calcium-sensitive potassium ion channel memristor of the entire connected Chay small-signal equivalent circuit in Figure 1b and Figure 16. For the readers’ convenience, the explicit formulas for computing the coefficients a

_{11}(Q

_{,K,Ca}), a

_{12}(Q

_{K}

_{,Ca}), b

_{11}(Q

_{,K,Ca}), b

_{12}(Q

_{,K,Ca}) and L

_{K}

_{,Ca}, R

_{1K,Ca}, R

_{2K,Ca}are summarized in Figure 15.

#### 5.4. Small-Signal Circuit Model of the Memristive Chay Model

_{I}= V

_{m}− E

_{I}, V

_{K}

_{,V}= V

_{m}− E

_{K}, and V

_{K}

_{,Ca}= V

_{m}− E

_{K}, respectively. Short-circuiting all the batteries, the equivalent small-signal circuit model of the third-order neuron circuit from Figure 1b about the operating point V

_{m}(Q) is found to be composed of one capacitor, two inductors, and six resistors, as shown in Figure 16. The local admittance Y(s; V

_{m}(Q)) of this linear circuit seen from the port and formed by the capacitor terminals about Q is given by

_{m}(Q) (resp. I), are also given in Figure 16 for readers’ convenience. We will cover the details of these regimes in the section on locally activity and edge of chaos. The circuit element R

_{1,I}is obtained by calculating the small signal model of the voltage-sensitive mixed ion channel nonlinear resistor from Figure 7 at equilibrium voltage V

_{m}(Q) = V

_{I}+ E

_{I}. Similarly, L

_{K}

_{,V}, R

_{1K,V}, and R

_{2K,V}are calculated from the small-signal equivalent circuit of the voltage-sensitive potassium ion channel memristor from Figure 11, and L

_{K}

_{,Ca}, R

_{1K,Ca}, and R

_{2K,Ca}are calculated from the small-signal equivalent circuit of the calcium-sensitive potassium ion channel memristor from Figure 15 at equilibrium voltage V

_{m}(Q), respectively. Note that V

_{K}

_{,V}+ E

_{K}, and V

_{K}

_{,Ca}+ E

_{K}must be replaced by V

_{m}(Q) in Figure 11 and Figure 15 by the small signal model of the voltage-sensitive potassium ion channel memristor and calcium-sensitive potassium ion channel memristor, respectively.

#### 5.4.1. Frequency Response

_{m}(Q)) by recasting Equation (80) into a rational function of the complex frequency variable s is as follows:

_{3}, b

_{2}, b

_{1}, b

_{0}, a

_{2}, a

_{1}, and a

_{0}are summarized in Figure 17.

_{m}(Q):

_{m}(Q)) and imaginary part ImY(iω; V

_{m}(Q)) from Equation (82) are given by,

_{m}(Q)) vs. ω, ImY(iω; V

_{m}(Q)) vs. ω, and the Nyquist plot ImY(iω; V

_{m}(Q)) vs. ReY(iω; V

_{m}(Q)) at the DC equilibrium voltage V

_{m}= −48.763 mV (resp., I = −66.671 μA), and V

_{m}= −27.984 mV (resp., I = 433.594 μA), respectively. Observe from Figure 18a,b that ReY(iω; V

_{m}(Q)) < 0, thereby confirming the memristive Chay model is locally active at each of the two operating points. Our extensive numerical computations show the two DC equilibria coincide with two Hopf bifurcation points, which are the origin of the oscillation, spikes, chaos, and bursting in excitable cells. We will discuss these two bifurcation points in the next section with a pole-zeros and eigen values diagram.

#### 5.4.2. Pole-Zero Diagram of the Small-Signal Admittance Function Y(s; V_{m}(Q)) and Eigen values of the Jacobian Matrix

_{m}(Q)) of Equation (81) is computed by factorizing its denominator and numerators as

_{m}(Q)) as a function of the voltage V

_{m}over −200 mV < V

_{m}< 200 mV is shown in Figure 19. Observe from Figure 19a,b that the two poles Re(p

_{1}), Re(p

_{2}) are negative while Im(p

_{1}), Im(p

_{2}) remain consistently zero for the specified DC input V

_{m}. This observation confirms that the two poles of the admittance function possess no complex frequencies.

_{i}) vs. the real part Re(z

_{i}) of the zeros as a function of the input voltage V

_{m}over the interval −55 mV ≤ V

_{m}≤ 25 mV. Observe that the real parts of the two zeros z

_{2}and z

_{3}are zero at V

_{m}= −48.763 mV (resp., I = −66.671 μA) and V

_{m}= −27.984 mV (resp., I = 433.594 μA), respectively. The corresponding points when Re(z

_{i}) = 0 are known as Hopf bifurcation points in bifurcation theory. Figure 20b,c show the zoomed version of Figure 20a near the two bifurcation points, respectively. It is also observed that the Re(z

_{2}) and Re(z

_{3}) lie in the open right half plane (RHP) between the bifurcation points −48.763 mV < V

_{m}< −27.984 mV (resp. −66.671 μA < I < 433.594 μA). Observe from Figure 21 that the eigen values, computed from the Jacobian matrix, associated with the ODEs (1)–(3) are identical to the zeros of the neuron local admittance Y(s; V

_{m}(Q)), as inferable from Figure 20, and expected from the Chua theory [3,4].

## 6. Local Activity, Edge of Chaos, and Hopf-Bifurcation in Memristive Chay Model

#### 6.1. Locally Active Regime

_{m}(Q)) lie in the open-right plane where Re(s

_{z}) > 0.

_{m}(Q)) has multiple zeros on the imaginary axis.

_{m}(Q)) has a simple zero on the imaginary axis s = iω

_{z}on the imaginary axis and ${K}_{Q}(i{\omega}_{z})\triangleq \underset{s\to i{\omega}_{z}}{\mathrm{lim}}=(s-i{\omega}_{z})Y(s;{V}_{m}(Q))$ is either a negative real number or a complex number.

_{m}(Q)) < 0 for some ω ϵ [−∞, +∞].

_{m}= −50 mV (resp. I = −74.316 μA) to V

_{m}= −23.5 mV (resp. I = 1.76 × 10

^{3}μA). Observe from Figure 22a the real part of the admittance of the frequency response ReY(iω; V

_{m}(Q)) > 0 at V

_{m}= −50 mV (resp. I = −74.316 μA), thereby confirming locally passive at this equilibrium point. However, when V

_{m}> −50 mV, our in-depth simulation in Figure 22b shows that ReY(iω; V

_{m}(Q)) = 0 at V

_{m}= −49.455 mV (resp. I = −70.919 μA) and Figure 22c,d show that ReY(iω; V

_{m}(Q)) < 0 at V

_{m}= −48.1 mV (resp. I = −62.681 μA) and V

_{m}= −26.5 mV (resp. I = 746.457 μA) respectively for some frequency ω, confirming an excitable cell is locally active at these equilibria. Our simulations in Figure 22e shows a further increase in the DC equilibrium voltage at V

_{m}= −24.685 mV (resp. I =1.291 × 10

^{3}μA), and the loci are tangential to the ω axis, i.e., ReY(iω; V

_{m}(Q)) = 0. However, when V

_{m}> −24.685 mV, say V

_{m}= −23.5 mV (resp. I = 1.76 × 10

^{3}μA), it is observed from Figure 22f that Re Y (iω; V

_{m}(Q)) > 0, and the memrisitve Chay model is no more locally active, confirming the cell is locally passive at this equilibrium. Therefore, the local activity regime that started above V

_{m}= −49.455 mV (resp. I = −70.919 μA) exists over the following regime

#### 6.2. Edge of Chaos Regime

_{m}(Q))(equivalent to the eigen values of the Jacobian matrix) lie in the open left-half plane, i.e., Re(z

_{p}) < 0 (eigen values λi < 0) as well as ReY(iω; V

_{m}(Q)) < 0. Figure 21a,b show the real part of the eigen values vanish at V

_{m}= −48.7631 mV (resp. I = −66.671 μA) with a pair of complex eigen values λ

_{2,3}= ±0.557i. It follows from the edge of chaos theorem that the corresponding equilibrium point is no longer asymptotically stable and becomes unstable thereafter, confirming the first edge of chaos regime over the following small interval:

_{2,3}= ±85.606i at DC equilibrium voltage V

_{m}= −27.984 mV (resp. I = 433.594 μA). It follows that the corresponding equilibrium point V

_{m}(Q) is no longer asymptotically stable below this equilibrium point, therefore confirming the existence of a second edge of chaos regime over the following interval:

_{m}(Q) to high equilibrium voltage V

_{m}(Q)).

#### 6.3. Hopf-Bifurcation

_{m}= −48.763 mV (resp. I = −66.671 μA) (The supercritical Hopf bifurcation points 1 and 2 observed in this paper are based on the numerical simulations and for the parameters listed in Table 2. The bifurcation phenomenon may vary for different parameters and environments.) shows that stimulus current I should be chosen within the very small edge of chaos domain 1, where the real part of the eigen values are negative. The result converges to DC equilibrium for any initial conditions. Likewise, if I is selected within bifurcation point 1, where the real part of the eigen values are positive, the result converges to a stable limit cycle. Therefore, it follows from the bifurcation theory that bifurcation point 1 is a super-critical Hopf bifurcation. Figure 23a,b show the numerical simulations at I = −68.118 μA and I = −65.077 μA, respectively. Observe from Figure 23a,b that I =−68.118 μA lying within the tiny subset of edge of chaos domain 1 converges to DC equilibrium and I = −65.077 μA lying in the open right half-plane (RHP) converges to spikes, respectively, confirming that bifurcation point 1 is a super-critical Hopf bifurcation.

_{m}= −27.984 mV (resp. I = 433.594 μA). The possibility of the above scenario is illustrated in Figure 24. Figure 24a shows the membrane potential V

_{m}converges to a stable DC equilibrium point when I = 440 μA, chosen within the edge of chaos domain 2. Figure 24b shows that when I = 430.884 μA, chosen very close and inside the bifurcation point 2, where the real part of the eigenvalue is positive and lies in the open right half plane (RHP), the transient waveform converges to a stable limit cycle as predicted by the Hopf super critical bifurcation theorem.

_{1}, λ

_{2}, and λ

_{3}) as a function of the DC stimulus current I (resp. membrane potential V

_{m}) at the DC equilibrium point Q. It is observed from Table 3 and Figure 21a,c that the two Hopf bifurcations points 1 and 2 occur at V

_{m}= −48.763 mV (resp. I = −66.671 μA) and V

_{m}= −27.984 mV (resp. I = 433.594 μA), respectively, where the eigen values are purely imaginary at these two equilibria. As I decreases (resp. V

_{m}decreases) from Hopf bifurcation point 1, the eigen values migrate to the left-hand side, confirming the real parts of the eigen values are no longer positive and thereby confirming the first negative real eigen values regime exists over the following interval.

_{m}increases) from the second bifurcation points, the positive real part of the eigen values migrates from the open right half to the open left half, confirming the second negative real eigen values regime over the following interval:

_{m}converging to a stable DC equilibrium at I = −90 µA, confirming the Hopf bifurcation theorem no longer holds at this equilibrium. Similarly, when DC stimulus currents I = −50 µA and −10 µA are chosen inside the two bifurcation points I = −66.671 μA and I = 433.594 µA, we observed different patterns of oscillations as shown in Figure 25b,c, confirming the bifurcation theorem holds in this regime. Likewise, when DC stimulus currents I = 10 μA and I = 2000 μA are applied within the bifurcation points I = −66.671 μA and I = 433.594, respectively, oscillation patterns emerge as depicted in Figure 26a,b. Similarly, Figure 26c illustrates the transient waveform of the membrane potential V

_{m}, indicating its convergence to a stable DC equilibrium at I = 500 μA. This observation suggests that the Hopf bifurcation theorem no longer holds at this equilibrium point.

_{KCa}of the calcium-sensitive potassium ion channel memristor is varied from 10 mS/cm

^{2}to 11.5 mS/cm

^{2}at stimulus current I = 0. Figure 27a shows the excitable membrane cell has a stable limit cycle with period one at g

_{K}

_{,Ca}= 10 mS/cm

^{2}. As the parameter g

_{K}

_{,Ca}increases to 10.7 mS/cm

^{2}, 10.75 mS/cm

^{2}, and 10.77 mS/cm

^{2}, the cell fires periods two, four, and eight, as shown in Figure 27b, Figure 27c, and Figure 28a, respectively. The change in period doubling is more apparent in calcium concentration (Ca) vs. time and V

_{m}vs. Ca, as shown at the bottom of Figure 27b, Figure 27c, and Figure 28a, respectively. Figure 28b shows the waveform of the memistive Chay model, confirming the existence of aperiodic oscillation (chaos) at g

_{K}

_{,Ca}= 11 mS/cm

^{2}. The firing of aperiodic oscillations from the cell can be clearly seen from the plot of Ca vs. time and V

_{m}vs. Ca in Figure 28b. A further increase in g

_{K}

_{,Ca}to 11.5 mS/cm

^{2}gives rise to the firing of the cell from aperiodic to rhythmic bursting, as shown in Figure 28c.

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Abbreviations of the Model Parameters

C_{m} | Membrane Capacitance |

E_{K} | Potential across K^{+} ion channel memristor |

E_{I} | Potential across mixed ion channel memristor |

E_{L} | Potential across leakage channel |

E_{Ca} | Potential across Ca^{2+} ion channel memristor |

g_{K,V} | Voltage-sensitive K^{+} ion-channel conductance |

g_{I} | Voltage-sensitive mixed ion channel conductance |

g_{L} | Leakage channel conductance |

g_{KCa} | Calcium activated potassium conductance |

k_{Ca} | Rate constant for the efflux of the intracellular Ca^{2+} ions |

ρ | Proportionality constant |

λ_{n} | Rate constant for K^{+} ion-channel opening |

m_{∞} | Probability of activation of the mixed ion channel in steady state |

α_{m} | The rate at which the activation of the mixed ion channel closed gates transition to an open state (s^{−1}) |

β_{m} | The rate at which the activation of the mixed ion channel open gates transition to the close state (s^{−1}) |

h_{∞} | Probability of inactivation of the mixed ion channel in steady state |

α_{h} | The rate at which the inactivation of the mixed ion channel closed gates transition to an open state (s^{−1}) |

β_{h} | The rate at which the inactivation of the mixed ion channel open gates transition to the close state (s^{−1}) |

n | Probability of n opening of the K^{+} ion channel memristor |

n_{∞} | Steady state value of n |

α_{n} | The rate at which K^{+} ion channel closed gates transition to an open state (s^{−1}) |

β_{n} | The rate at which K^{+} ion channel opened gates transition to an close state (s^{−1}) |

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**Figure 1.**Typical Chay neuron model of an excitable cell [30]. (

**a**) Electrical circuit model, following conventional assumption as time varying conductances [1]. (

**b**) Equivalent memristive Chay model based on Chua’s memristive theory [2,3,4]. The potential E

_{Ca}for Ca

^{2+}ion given in the rate of the calcium concentration in Equation (3) is not an external battery source and not shown in external (

**a**) and (

**b**), respectively.

**Figure 2.**Output waveform plotted on the i

_{I}vs. v

_{I}plane when the input voltage v

_{I}= 100sin(2πft) mV is applied with three different frequencies, namely f = 100 Hz, 200 Hz, and 1 KHz, to the voltage-sensitive mixed ion channel. The output nonlinear waveform observed in Figure 2 for different frequencies confirms the mixed ion channel is a nonlinear resistor.

**Figure 3.**Pinched hysteresis loops of voltage-sensitive potassium ion channel memristor at frequencies f = 100 KHz, 500 KHz, and 4 MHz for the input signal v

_{K}

_{,V}(t) = 100sin(2πft) mV.

**Figure 4.**Pinched hysteresis loops of the calcium-sensitive potassium ion channel memristor at frequencies f = 10 Hz, 30 Hz, and 200 Hz for the input signal v

_{K}

_{,Ca}(t) = 100sin(2πft) mV.

**Figure 5.**(

**a**) Memristive DC Chay model at equilibrium voltage V

_{m}. (

**b**) DC V-I curve of a mixed ion channel nonlinear resistor at equilibrium voltage V

_{I}= V

_{m}− E

_{I}. (

**c**) DC V-I curve of voltage-sensitive potassium ion channel memristor at equilibrium voltage V

_{K}

_{,V}= V

_{m}− E

_{K}. (

**d**) DC V-I curve of the calcium-sensitive potassium ion channel memristor at equilibrium voltage V

_{K}

_{,Ca}= V

_{m}− E

_{K}. (

**e**) DC V-I curve of the leakage channel at equilibrium voltage V

_{L}= V

_{m}− EL. (

**f**) Plot of the DC V-I curve of the memristive Chay model at membrane voltage V

_{m}.

**Figure 6.**(

**a**) Small-signal circuit model of the voltage-sensitive mixed ion channel nonlinear resistor about the DC equilibrium point Q

_{I}(V

_{I}, I

_{I}). (

**b**) Plot of the coefficient a

_{12}and resistance R

_{1,I}as a function of the DC equilibrium voltage V

_{I}= V

_{m}− E

_{I}, where E

_{I}= 100 mV. R

_{1,I}< 0 over the range of local activity, edge of chaos 1 and edge of chaos 2 of the mixed ion channel nonlinear resistor is identified with respect to V

_{I}of the entire Chay circuit in Figure 1b and Figure 16.

**Figure 7.**Explicit formulas for computing the coefficients a

_{12}(Q

_{I}) of the voltage-sensitive mixed ion channel nonlinear resistor.

**Figure 8.**Small-signal equivalent circuit model of the voltage-sensitive potassium ion channel memristor about the DC equilibrium point Q

_{K}

_{,V}(V

_{K}

_{,V}, I

_{K}

_{,V}).

**Figure 9.**Plot of coefficients (

**a**) a

_{11}, (

**b**) a

_{12}, (

**c**) b

_{11}, and (

**d**) b

_{12}of the voltage-sensitive potassium ion channel memristor as a function of the DC equilibrium voltage V

_{K}

_{,V}.

**Figure 10.**(

**a**) Inductance L

_{K}

_{,V}(

**b**) resistance R

_{1K,V}and (

**c**) resistance R

_{2K,V}of the voltage-sensitive potassium ion channel memristor as a function of DC equilibrium voltage V

_{K}

_{,V}= V

_{m}− E

_{K}where E

_{K}= −75 mV. L

_{K}

_{,V}> 0, R

_{1KV}> 0 and R

_{2KV}> 0 shown in figures over the local activity, edge of chaos 1 and edge of chaos 2 are just corresponding range of the voltage with respect to V

_{K}

_{,V}of the entire connected Chay small signal equivalent circuit of Figure 1b and Figure 16.

**Figure 11.**Explicit formulas for computing the coefficients a

_{11}(Q

_{K}

_{,V}), a

_{12}(Q

_{K}

_{,V}), b

_{11}(Q

_{K}

_{,V}), b

_{12}(Q

_{K}

_{,V}) and L

_{K}

_{,V}, R

_{1K,V}, R

_{2K,V}of the voltage-sensitive potassium ion channel memristor.

**Figure 12.**Small-signal equivalent circuit model of the calcium-sensitive potassium ion channel memristor about the DC equilibrium point Q

_{K}

_{,Ca}(V

_{K}

_{,Ca}, I

_{K}

_{,Ca}).

**Figure 13.**Plot of coefficients (

**a**) a

_{11}(

**b**) a

_{12}, (

**c**) b

_{11}, and (

**d**) b

_{12}of the calcium-sensitive potassium ion channel memristor as a function of the DC equilibrium voltage V

_{K}

_{,Ca}.

**Figure 14.**(

**a**) Inductance L

_{K}

_{,Ca}(

**b**) resistance R

_{1K,Ca}and (

**c**) resistance R

_{2K,Ca}of the calcium-sensitive potassium ion channel memristor as a function of DC equilibrium voltage V

_{K}

_{,Ca}. L

_{K,Ca}> 0, R

_{1K,Ca}> 0, and R

_{2KCa}> 0 over the edge of chaos 1 and edge of chaos 2 with respect to V

_{K}

_{,Ca}of the entire connected Chay small-signal equivalent circuit of Figure 1b and Figure 16.

**Figure 15.**Explicit formulas for computing the coefficients a

_{11}(Q

_{K}

_{,Ca}), a

_{12}(Q

_{K}

_{,Ca}), b

_{11}(Q

_{K}

_{,Ca}), b

_{12}(Q

_{K}

_{,Ca}) and L

_{K}

_{,Ca}, R

_{1K,Ca}, R

_{2K,Ca}of the calcium-sensitive potassium ion channel memristor.

**Figure 16.**Small-signal equivalent circuit model of the memristive Chay model. The DC equilibrium voltage V

_{m}is computed at V

_{m}= V

_{I}+ E

_{I}for mixed ion channel non-linear resistor, V

_{m}= V

_{K}

_{,V}+ E

_{K}for voltage-sensitive potassium ion channel memristor, and V

_{m}= V

_{K}

_{,Ca}+ E

_{K}for calcium-sensitive potassium ion channel memristor, respectively.

**Figure 18.**Small-signal admittance frequency response and Nyquist plot of the memristive Chay neuron model at (

**a**) V

_{m}= −48.763 mV (resp., I = −66.671 μA) and (

**b**) V

_{m}= −27.984 mV (resp., I = 433.594 μA). Observe that ReY(iω; V

_{m}(Q)) < 0 at the two Hopf-bifurcation points.

**Figure 19.**Poles diagram of the small-signal admittance function Y(s; V

_{m}(Q)) as a function of V

_{m}over −200 mV < V

_{m}< 200 mV (

**a**) The top and bottom figures are the plots of the real part of pole 1 Re(p

_{1}) and the imaginary part of pole 1 Im(p

_{1}), respectively. (

**b**) The top and bottom figures are the plots of the real part of pole 2 Re(p

_{2}) and the imaginary part of pole 2 Im(p2), respectively.

**Figure 20.**Zeros diagram of the small-signal admittance function Y(s; V

_{m}(Q)) (

**a**) Nyquist plot of the zeros z

_{1}, z

_{2}, z

_{3}in Im(z

_{i}) vs. Re(z

_{i}) plane (

**b**) Nyquist plot near the Hopf-bifurcation point 1, V

_{m}= −48.763 mV (resp., I = −66.671 μA). (

**c**) Nyquist plot near the Hopf-bifurcation point 2, V

_{m}= −27.984 mV (resp., I = 433.594 μA).

**Figure 21.**Plot of the loci of the eigen values of the Jacobian Matrix (

**a**) Nyquist plot of the eigen values λ

_{1}, λ

_{2}, λ

_{3}in Im(λ

_{i}) vs. Re(λ

_{i}) plane. (

**b**) Nyquist plot near the Hopf-bifurcation point 1, V

_{m}= −48.763 mV (resp., I = −66.671 μA). (

**c**) Nyquist plot near the Hopf-bifurcation point 2, V

_{m}= −27.984 mV (resp., I = 433.594 μA). Our numerical computations confirm the zeros of the admittance functions Y(s; V

_{m}(Q)) obtained in Figure 20 are identical to the eigen values of the Jacobian matrix.

**Figure 22.**Plot of Re(iω; V

_{m}(Q)) to illustrate the local activity principle at (

**a**) V

_{m}= −50 mV (resp. I = −74.316 μA) (

**b**) V

_{m}= −49.455 mV (resp. I = −70.919 μA), (

**c**) V

_{m}= −48.1 mV (resp. I = −62.681 μA), (

**d**) V

_{m}= −26.5 mV (resp. I = 746.457 μA), (

**e**) V

_{m}= −24.685 mV (resp. I = 1.291 × 10

^{3}μA), (

**f**) V

_{m}= −23.5 mV (resp. I = 1.76 × 10

^{3}μA), respectively.

**Figure 23.**Numerical simulations to confirm the super-critical Hopf bifurcation at bifurcation point 1. Plot of membrane potential V

_{m}at (

**a**) I = −68.118 μA which lies inside the tiny subset of edge of chaos domain 1 and beyond bifurcation point 1 converges to the DC equilibrium; (

**b**) I = −65.077 μA, chosen just to the right of bifurcation point 1, where the real parts of two zeros of the neuron local admittance lie on the open right half plane (RHP), converges to the spikes.

**Figure 24.**Numerical simulations to confirm the super-critical Hopf bifurcation at bifurcation point 2. (

**a**) Plot of membrane potential V

_{m}which converges to stable DC equilibrium when I = 440 µA chosen inside the tiny subset of edge of chaos domain 2 and near and beyond the bifurcation point 2. (

**b**) Membrane potential converging to oscillation as predicted by the Hopf bifurcation theorem when I = 430.884 µA is chosen inside the bifurcation point (open right-half pane).

**Figure 25.**Patterns of oscillations when stimulus current I is chosen beyond and inside the bifurcation points. (

**a**) DC pattern observed when I = −90 µA chosen beyond bifurcation point 1 (I = −66.671 μA). Different patterns of oscillations occur when I is chosen between the two bifurcation points I = −66.671 μA and I = 433.594 μA at (

**b**) I = −50 μA and (

**c**) I = −10 μA.

**Figure 26.**Patterns of oscillations when stimulus current I is chosen inside and beyond the bifurcation points. Oscillation patterns when I is chosen between the two bifurcation points I = −66.671 μA and I = 433.594 μA, at (

**a**) I = 10 μA, and (

**b**) I = 200 μA. (

**c**) The DC pattern when I = 500 μA is chosen beyond bifurcation point 2 (I = 433.594 μA).

**Figure 27.**Different patterns of oscillations when g

_{KCa}varied from 10 mS/cm

^{2}to 10.75 mS/cm

^{2}at DC situmulus current I = 0. (

**a**) Period-1 oscillation at g

_{K}

_{,Ca}= 10 mS/cm

^{2}(

**b**) Period-2 oscillation at g

_{K}

_{,Ca}= 10.7 mS/cm

^{2}(

**c**) Period-4 oscillation at g

_{K}

_{,Ca}= 10.75 mS/cm

^{2}. The simulations were performed at the initial conditions V

_{m}(0) = −50 mV, n(0) = 0.1, and Ca(0) = 0.48.

**Figure 28.**Different patterns of oscillations when g

_{KCa}varied from 10.77 mS/cm

^{2}to 11.5 mS/cm

^{2}at DC stimulus current I = 0. (

**a**) Period-8 oscillation at g

_{K}

_{,Ca}= 10.77 mS/cm

^{2}; (

**b**) Aperiodic (chaotic) oscillation at g

_{K}

_{,Ca}= 11 mS/cm

^{2}; (

**c**) Bursting at g

_{K}

_{,Ca}= 11.5 mS/cm

^{2}. The simulations were performed at the initial conditions V

_{m}(0) = −50 mV, n(0) = 0.1, and Ca(0) = 0.48.

C_{m} | 1 mF/cm^{2} | g_{K}_{,V} | 1700 mS/cm^{2} |

E_{K} | −75 mV | g_{I} | 1800 mS/cm^{2} |

E_{I} | 100 mV | g_{L} | 7 mS/cm^{2} |

E_{L} | −40 mV | g_{K}_{,Ca} | 10 mS/cm^{2} |

E_{Ca} | 100 mV | K_{ca} | 3.3/18 mV |

λ_{n} | 230 | ρ | 0.27 |

Models | Memristive Models | Strengths | Limitations |
---|---|---|---|

HH [1] | The potassium ion channel and sodium ion channel in the HH model are represented by generic memristors [3]. | It is a framework to understand the emergence of action potential propagation in neurons based on the experimental data of the squid giant axon. | It is difficult to generalize to all neurons. Incapable of producing complicated bursting patterns |

FitzHugh–Nagumo [31] | It does not follow state-dependent Ohm’s law and cannot model with memristors. | Simplified model of neuronal excitation. | Not accurately represent all neuronal behaviors. Incapable of producing bursting |

ML [8] | It was modeled that the state-independent (dependent) calcium ion channel acts as a nonlinear resistor (generic memristor) and state-dependent potassium ion channel acts as a generic memristor [9,10]. | It is initially presented a model for the barnacle muscle fiber, and later it was considered a popular and simplified representation of the neuron model. | Limited in capturing certain neuronal dynamics. Cannot produce bursting patterns. |

Chay [30] | We are proposing a framework that the cells of excitable membranes can be modeled as the networks of memristors. | Novel model of excitable cells to capture multiple neuronal states, such as action potentials, periodic oscillations, aperiodic oscillations, spikes, and bursting patterns. | Limited validation in experimental contexts and a lack of details for some applications. |

**Table 3.**Computation of the potassium ion channel activation n, calcium concentration Ca, and eigen values (λ

_{1}, λ

_{2}, and λ

_{3}) as a function of the stimulus current I (resp. membrane potential V

_{m}). Rows 5 to 7 pertain to the edge of chaos 1, rows 8 to 16 pertain to the unstable local activity domain, and rows 17 to 20 pertain to the edge of chaos 2. Rows 7 and 17 pertain to Hopf bifurcation points 1 and 2, respectively, for the memristive Chay neuron model.

S.N | V_{m} (mV) | I (µA) | n | Ca | λ_{1} | λ_{2} | λ_{3} |
---|---|---|---|---|---|---|---|

1. | −52.00 | −87.02 | 0.08 | 0.04 | −40.515 | −3.842 | −0.084 |

2. | −51.00 | −80.63 | 0.08 | 0.05 | −40.107 | −2.871 | −0.111 |

3. | −50.50 | −77.46 | 0.09 | 0.06 | −39.891 | −2.289 | −0.139 |

4. | −50.00 | −74.32 | 0.09 | 0.07 | 39.666 | −1.617 | −0.196 |

5. | −49.455 | −70.919 | 0.94 | 0.08 | −39.408 | −0.533 − 0.174i | −0.533 + 0.174i |

6. | −49.00 | −68.12 | 0.1 | 0.1 | −39.181 | −0.19 − 0.525i | −0.19 + 0.525i |

7. | −48.763 | −66.671 | 0.1 | 0.1 | −39.058 | 0 − 0.557i | 0 + 0.557i |

8. | −48.50 | −65.08 | 0.1 | 0.11 | −38.917 | 0.222 − 0.51i | 0.2215 + 0.5097i |

9. | −46.00 | −51.02 | 0.12 | 0.21 | −37.32 | 0.046 | 5.736 |

10. | −45.00 | −46.37 | 0.13 | 0.27 | −36.512 | 0.027 | 8.498 |

11. | −42.00 | −39.37 | 0.16 | 0.53 | −33.218 | 0.0001 | 18.604 |

12. | −40.00 | −42.78 | 0.18 | 0.79 | −29.899 | −0.0084 | 25.99 |

13. | −38.00 | −51.26 | 0.21 | 1.13 | −24.898 | −0.0112 | 31.992 |

14. | −32.00 | 17.59 | 0.29 | 2.57 | −0.061 | 11.669 − 38.01i | 11.669 + 38.01i |

15. | −30.00 | 160.68 | 0.32 | 3.12 | −0.053 | 8.049 − 61.778i | 8.049 + 61.778i |

16. | −28.00 | 430.84 | 0.35 | 3.65 | −0.051 | 0.08 − 85.421i | 0.08 + 85.421i |

17. | −27.984 | 433.594 | 0.35 | 3.65 | −0.051 | 0 − 85.606i | 0 + 85.606i |

18. | −27.00 | 628.91 | 0.36 | 3.89 | −5.556 − 97.197i | −5.556 + 97.197i | −0.051 |

19. | −25.50 | 1.02 × 10^{3} | 0.39 | 4.22 | −15.942 − 114.607i | −15.942 + 114.607i | −0.0501 |

20. | −24.685 | 1.291 × 10^{3} | 0.40 | 4.37 | −22.466 − 123.858i | −22.466 + 123.858i | −0.0499 |

21. | −23.00 | 1.99 × 10^{3} | 0.43 | 4.64 | −37.643 − 142.384i | −37.643 + 142.384i | −0.0497 |

22. | −22.00 | 2.5 × 10^{3} | 0.44 | 4.75 | −47.529 − 152.923i | −47.529 + 152.923i | −0.0496 |

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**MDPI and ACS Style**

Sah, M.; Ascoli, A.; Tetzlaff, R.; Rajamani, V.; Budhathoki, R.K.
Modeling Excitable Cells with Memristors. *J. Low Power Electron. Appl.* **2024**, *14*, 31.
https://doi.org/10.3390/jlpea14020031

**AMA Style**

Sah M, Ascoli A, Tetzlaff R, Rajamani V, Budhathoki RK.
Modeling Excitable Cells with Memristors. *Journal of Low Power Electronics and Applications*. 2024; 14(2):31.
https://doi.org/10.3390/jlpea14020031

**Chicago/Turabian Style**

Sah, Maheshwar, Alon Ascoli, Ronald Tetzlaff, Vetriveeran Rajamani, and Ram Kaji Budhathoki.
2024. "Modeling Excitable Cells with Memristors" *Journal of Low Power Electronics and Applications* 14, no. 2: 31.
https://doi.org/10.3390/jlpea14020031