Keywords

1 Introduction

Cable equation and its implementation in modeling and simulation of neuronal signal processing have been playing significant role. Its implication is found in modeling of either signal propagation in a neuronal fiber or extracellular field potential models thus giving theoretical insight into complex neuronal processes. The technical understanding of varying action potential shape and change in velocity in a neuronal fiber of varying diameter has been put forward in [6, 15, 17] using cable equation approach, which results in Rall’s equivalent cylinder and Rall’s 3/2 rule, implementable in branching dendritic fibers for a class of neuronal morphology. Cable model in [19] combined with Rall’s 3/2 rule and Agmon-Snir [1] extension model of cable equation shows its implementation in axonal regions and passive dendrites. On similar notes, in [4, 10, 13, 16, 18, 19] cable models found its uses in understanding of different dendritic morphology and structures. Even extended models of the infamous cable equation have found its place in modeling effects of local field potentials (LFP) and their respective effects in neuronal activities and cognition [3].

Numerous research has been put forward in order to understand the role of complex neuronal arbors, either in understanding their role in signal processing or in the understanding of LPF dynamics due to complex structures. In [9], incorporating ephaptic interactions between neuronal fibers, Holts et al. modeled ephaptic depolarization and discussed their probable functional relevance. In literature [11], the neuronal signal noises are interpreted due to channel properties, thermal noise of membrane and random background activity, considering the fluctuation of membrane conductance to be significantly lower than the resting conductance. In Hasselmo et al. [8], it has been found from extracellular recordings and computational modeling that the sub-threshold membrane potential interaction with different branches plays important role in theta frequency oscillations whereas Anastassiou [2] discusses the capability of ephaptic coupling of extracellular field and its role in low-frequency oscillations in cortical neuron. Similarly, literature [7, 20] also discusses the capability of extracellular field in mediating blocked action potential, correlates the extracellular potential as one of the important aspects of neuronal signal processing [12]. Such models of sustained cortical oscillations and ephaptic coupling of neuronal fibers have influenced to further investigate the cause and effects of the extracellular potential on neuronal signal processing. An attempt is made to model and understands the effect of extracellular potential on nerve membrane using a detailed cable model. In the methods section, representation of the cable model corresponding to a pair of passive dendritic fiber has been discussed and equivalent membrane potential equations are designed whereas, in the results section, the simulated results of the membrane equations are discussed.

2 Methods: Interference Model and Its Cable Representation

Figure 1 is the representation of two parallel dendritic fibres where \(Ra_1\) and \(Ra_2\) are the axial endoplasmic resistance, Re is the exoplasmic resistance, \(Ri_1\) and \(Ri_2\) are the membrane resistance and \(C_m\) is the membrane capacitance. The corresponding membrane constants are considered for a length of \(\varDelta x\) as seen in Fig. 1.

Fig. 1.
figure 1

(a) Model of passive dendritic fiber pairs. (b) Equivalent cable representation circuit considering endoplasmic resistance, exoplasmic resistance, membrane resistance, membrane capacitance, and voltage and current effects in the circuit due to interference between the two fibers during signal transmission/propagation.

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Considering Fig. 1, the node potential for the external and internal section of the fiber is represented as \(Vi_1\), \(Vi_2\), Ve and \(V_{m1}\), \(V_{m2}\) being the membrane potential of the two fibers, applying Kirchhoff’s voltage and current law at the system circuit gives

$$\begin{aligned} \frac{dV_{i1}}{dx}=-Ri_1 Ia_1,~\frac{dV_{i2}}{dx}=-Ri_2 Ia_2,~\frac{dV_{e}}{dx}=Re Ie \end{aligned}$$
(1)
$$\begin{aligned} I_{T1}=\frac{dIa_1}{dx},~I_{T2}=\frac{dIa_2}{dx},~I_e =I_{T1}+I_{T2} \end{aligned}$$
(2)

Using Eqs. 1 and 2, the membrane potential for individual fibers are given as

$$\begin{aligned} \frac{d^2V_{m1}}{dx^2}=Ra_1 Cm_1 \frac{dV_{m1}}{dt}+\left( Ra_1 I_{ionic1}+Re I_{ionic2}\right) \end{aligned}$$
(3)
$$\begin{aligned} \frac{d^2V_{m2}}{dx^2}=Ra_2 Cm_2 \frac{dV_{m2}}{dt}+\left( Ra_2 I_{ionic2}+Re I_{ionic1}\right) \end{aligned}$$
(4)

Simplifying Eqs. 3 and 4, considering the fiber to be equipotential over small length and equating for time response of the system of equation for a definite length of the dendritic fibers as a function of membrane constants, the system of equation can be described as

$$\begin{aligned} \frac{dV_{m1}}{dt}=\frac{Re}{\left( Ra_1Ra_2-Re^2\right) Cm_1}\left( Ra_2I_{ionic2}+ReI_{ionic1}\right) \end{aligned}$$
(5)
$$\begin{aligned} \frac{dV_{m2}}{dt}=\frac{Re}{\left( Ra_1Ra_2-Re^2\right) Cm_2}\left( Ra_1I_{ionic1}+ReI_{ionic2}\right) \end{aligned}$$
(6)

where \(I_{ionic1}\) and \(I_{ionic2}\) are ionic currents due to membrane leakage or membrane transport.

Considering the two dendritic fibers being triggered due to active membrane on injecting two currents \(I_{in1}\), \(I_{in2}\) on either of the fiber, the passive propagating response of the system is given as

$$\begin{aligned} \frac{dV_{m1}}{dt}=I_{inj1}+\frac{Re}{\left( Ra_1Ra_2-Re^2\right) Cm_1}\left( Ra_2I_{ionic2}+ReI_{ionic1}\right) \end{aligned}$$
(7)
$$\begin{aligned} \frac{dV_{m2}}{dt}=I_{inj2}+\frac{Re}{\left( Ra_1Ra_2-Re^2\right) Cm_2}\left( Ra_1I_{ionic1}+ReI_{ionic2}\right) \end{aligned}$$
(8)
Fig. 2.
figure 2

Interference responses from the model.

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3 Results and Discussion

All results and data are generated in the XPP solver for solving differential equations. XPP solver is basically a differential equation solver, initially designed and written by Bard Ermentrout and John Rinzel to simulate differential equations associated with nerve membrane dynamics and phaseplane analysis. Current version incorporates number of solvers that can solve differential equations, delay equations stochastic equations etc. and supports handling upto 590 differential equations. The membrane parameters are considered from literature [5]. Figure 2(a) is the transient response of the interference model of two parallel dendritic fibres and Figs. 2(b), 3(a) and (b) are the responses of interference model implementing two dendritic fibres of same diameters facilitating spike train propagation.

Fig. 3.
figure 3

Interference responses from the model on baseline shift and baseline wondering.

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Figure 2(a) is the transient response of the system of equations representing the response of a passive fiber due to potential change in a nearby fiber. As a dendritic fiber stabilizes from a potential of 0 mV to its resting potential, the membrane potential of a nearby fiber, initially at rest, is perturbed due to interference from the first fiber. Figure 2(b) is the model response considering each fiber (\(2\,\upmu \mathrm{m}~diameter\)) facilitating spike train propagation that results in inter-fiber interference. Figure 2(b) shows the interference response of fiber 1 due to signal propagation in fiber 2, which results in small noise like structures convolving with the original signal, which suggests that the high-frequency noise in the propagating signal might be the result of interference from another nearby fiber facilitating signal transmission at same instance of time.

Fig. 4.
figure 4

Interference responses from the model showing amplification of the propagating signal.

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Also, Fig. 3(a) shows the model response of fiber 2 due to spike train propagation in fiber 1, where single fiber facilitates spike train propagation. Due to spike train propagation in fiber 1, sustained oscillations are produced in the fiber 2 which was initially devoid of any spiking activity. Again, Fig. 3(b) is the response of interference model, considering two fiber of diameter \(5~\upmu \)m each. This response model shows significant shift in the baseline of the propagating signal along with minor baseline wondering as discussed in [14], which suggests that the baseline potential shift and baseline wondering might also be the consensus result due to propagating signal in nearby fiber when interference surface exposed is over a longer distance.

Figure 4(a) and (b) are the responses of two fibers (\(2~\upmu \)m diameter each) showing the effect of on phase and off phase signals on the interference model, considering the fibers to overlay over smaller length. The response shows the on phase signals affecting very less in the baseline potential shift of the two fibers whereas, as the phase difference between the two signals gradually increases, the baseline wandering and shifts get more prominent. Apart from the baseline shift and baseline wondering, the model seems to amplify the propagating signal as the phase shift between the two signals in the nearby fibers increases.

4 Conclusion

As the endoplasm of a neuron is more negatively charged than the exoplasm, the membrane potential of a neuron is negative. During the application of stimulation, the ion density of the inside with respect to the outside of the membrane changes significantly that gives rise to the action potential. Due to such dynamics of the cytoplasm, the nearby fibers also get affected and the dynamic behavior of the nearby fibers also changes. From the results in Fig. 2(a), it can be seen that as a fiber progressively stabilizes to its resting state by exchanging \(Na^+\) and \(K^+\) ions with the exoplasm, the instantaneous change in exoplasmic ion concentration starts ionic motion to stabilize the system which results in attraction of new negative ions to balance the ion exchange. This, in turn, results in interference effect in the other nearby fiber in terms of baseline potential shift for small instance of time due to instantaneous perturbation of extracellular ion concentration within the vicinity. Similar effects can be seen in Figs. 2(b), 3(a), (b) etc. which has some additional effects such as interference noise, signal amplification and baseline wondering. Results in Fig. 4(a) and (b) also suggests that the amplification and baseline shift is minimal when two fibers spikes are at same phase whereas the amplification and baseline shift increases with increase in phase shift between the two respective signals.