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Edited by: Abdelmalik Moujahid, University of the Basque Country, Spain

Reviewed by: Jun Ma, Lanzhou University of Technology, China; Daya Shankar Gupta, Camden County College, United States

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Electrical excitation of neural tissue has wide applications, but how electrical stimulation interacts with neural tissue remains to be elucidated. Here, we propose a new theory, named the Circuit-Probability theory, to reveal how this physical interaction happen. The relation between the electrical stimulation input and the neural response can be theoretically calculated. We show that many empirical models, including strength-duration relationship and linear-non-linear-Poisson model, can be theoretically explained, derived, and amended using our theory. Furthermore, this theory can explain the complex non-linear and resonant phenomena and fit

Neuromodulation by electrical stimulation has proven itself as an effective treatment for medical conditions in many therapeutic situations, including deep brain stimulation [e.g., Parkinson's disease; Shah et al.,

Here, we propose a new theory, named Circuit-Probability (C-P) theory, to provide a physical framework, which is completely different from the conventional way of using H-H model with E-field modeling. Then, we show that some widely-used empirical models and rules can be intuitively derived from the C-P theory.

How should we analyze tissue response to an external stimulation? To answer this question, we performed a thought experiment, which ultimately led to our new framework of Circuit-Probability. When considering the electrode-tissue interaction, the first question is how the electrode is bridged to the tissue. We know the activation of action potential is induced by the gating of the voltage-dependent ion channels. Then, for electrode-tissue interaction, the key issue is how the electrical input affects the voltage on these ion channels. Considering the cell membrane is a capacitor, which is impermeable to ions, it affects its electrical response in two aspects. Firstly, the voltage changes on the capacitor, which is induced by charging and discharging procedures, will generate a different waveform in response to the input waveform. And the charging and discharging procedures are not only affected by the capacitor itself, but also affected by its peripheral circuit. Secondly, the E-field will always be perpendicular to the plate of the capacitor, which is the cell membrane surface, and the direction of the E-field is only determined by the orientation of the capacitor. Apparently, the correct voltage waveform and correct E-field direction can be both obtained with a proper circuit involving the capacitor of cell membrane. This is why we use a circuit to characterize the electric response on the cell membrane.

With a proper circuit, we can model the voltage waveform. Then, from this voltage waveform, how can we know the stimulation strength? In the

Up to here, we have obtained a basic framework of Circuit-Probability based on pure physical reasoning. The proper circuit configuration can only be fitted using experimental results, which is a posteriori, while the probability equation can obtained by theoretical derivation, which is a priori.

Here we firstly show how to theoretically derive the probability equation.

In the electrical stimulation of a neuron, we assume that electron transition of the protein causes the opening of sodium ion channel, which then generates an action potential (AP). Electron transition is a quantum phenomenon, which is random. Hence, the generation of APs can be described with an exponential distribution for quantum event:

Here

then the normal exponential distribution is the special form when λ(

Meanwhile,

We have three electrophysiological considerations for

_{Threshold}. In this condition, the AP cannot be generated.

_{Threshold}|, when _{Threshold}.

_{Threshold}| goes to infinite. So

With these three considerations, one possible form of

The equation can be re-written as:

Here, α, β,

To simplify the equation, here we assume that

Then the complete expression of λ is:

Considering the voltage waveform,

In this equation, α, β, and _{Threshold} are three parameters to be determined by data fitting.

For a specific voltage waveform as shown in

where _{λ} is the area of the λ waveform. A detailed analysis of the probability calculus can be found in

Parameter illustration of the probability calculus. _{λ}.

Then, we build a proper circuit using the results shown in

Illustration of the Circuit-Probability (C-P) theory with experiment and modeling results from the Common Peroneal (CP) nerve stimulation with sine-wave current.

The shapes of these four curves are quite different, showing a complicated changing trend with increasing current amplitude. For the curves of 20 and 40 μA, a clear resonance effect can be observed. However, 80 μA curve shows an initial decline, before increasing to a resonance frequency. The curve of 200 μA shows a monotonically decreasing trend without the resonance effect. Despite these variations, C-P theory can still reproduce the general shapes of the curves via probability mapping (

Modeling parameters.

_{1}(Ω) |
_{2}(Ω) |
_{3}(Ω) |
_{Threshold}(V) |
|||||
---|---|---|---|---|---|---|---|---|

1 (d&e) | 3,45,000 | 5,000 | 10,000 | 9n | 1.9545 | 2,000 | 0.1 | −0.6 |

3 (b,c) | 16,579 | 100 | 3,000 | 12n | 2.1109 | NA | NA | From −0.09 to −0.17 |

S4.1 (b) | 16,579 | 100 | 3,000 | 12n | 2.1109 | 1,200 | 0.01 | −0.08 |

S4.2 (a) | 11,052 | 100 | 3,000 | 12n | 0.5277 | 2,000 | 0.04 | −0.048 |

S5 (a-i) | 5,181 | 100 | 200 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S5 (a-ii) | 5,181 | 100 | 2,000 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S5 (b-i) | 10,362 | 100 | 200 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S5 (b-ii) | 10,362 | 100 | 2,000 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S5 (c-i) | 20,723 | 100 | 200 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S5 (c-ii) | 20,723 | 100 | 2,000 | 12n | 0.1938 | 2,000 | 0.1 | −0.1 |

S6.2.1 (a) | 80,000 | 300 | 1,700 | 18n | 0.1086 | NA | NA | NA |

S6.2.1 (b) | 2,656 | 1,800 | 800 | 18n | 0.0813 | NA | NA | NA |

S6.2.1 (c) | 2,656 | 1,800 | 800 | 18n | 0.0813 | NA | NA | NA |

S6.2.1 (d) | 2,000 | 1,350 | 500 | 10n | 0.1464 | NA | NA | NA |

S6.2.1 (e) | 3,701 | 350 | 500 | 10n | 0.1464 | NA | NA | NA |

S6.2.1 (f) | 9,000 | 1,350 | 500 | 10n | 0.2326 | NA | NA | NA |

S6.3.1.1 (b) | 60,286 | 1,800 | 2,000 | 4n | 5.2335 | 600 | 0.8 | −0.7 |

S6.3.1.2 (b) | 72,343 | 4,600 | 14,400 | 4n | 5.2335 | 1,500 | 0.06 | −9.69 |

S6.3.1.3 (b) | 100 | 100 | 300 | 100n | 0.1621 | 45,000 | 0.0075 | −0.006 |

S6.4.1 (b) | 12,384 | 1,200 | 18,000 | 10n | 4.9687 | 13,000 | 0.5 | −0.35 |

S6.4.2 (b) | 5,000 | 30 | 200 | C1 = 400n; C2 = 5,000n | 0.0702 | 2,000 | 0.015 | −0.009 |

S6.5.1 (b) | 90,000 | 100 | 600 | 12n | 0.1629 | 17,000 | 0.58 | −0.22 |

The C-P framework and the probability calculus equation is achieved by reasoning, which is a priori, rather than a posteriori. This is very unusual for biological research. Meanwhile, the circuit is still of a preliminary configuration. To validate the correctness of this priori theory, a series of experiments on four types of non-neural and neural tissues using a rat model were conducted: the skeletal muscles (

Meanwhile, C-P theory can give a unique prediction: the electrical voltage response by electrical stimulation, which is conventionally considered as the stimulus artifact, can be well-fitted by the voltage response of the circuit in

This C-P theory provides a physical understanding of the electrical nerve stimulation, which is not available in previous theories and models. Thus, most of the phenomenological models and theories can be directly derived or even amended from C-P theory. Here we just show how to derive and correct two well-known phenomenological models in electrical nerve stimulation: strength-duration relationship (Lapicque,

Firstly, we will derive and amend the strength-duration relationship. Previously, it is widely believed that charge is the factor to induce nerve stimulation. In such charge based theory, there is an empirical linear relationship between the threshold charge level and pulse duration, which is called Weiss's strength–duration equation (Weiss,

where _{rh} is the rheobase current, _{ch} is the chronaxie, and

Apparently, these two equations are just mathematical descriptions without explaining how _{rh} happen and why the curve follows a specific trend.

As follows is the derivation of this relationship with physical definition of _{rh}.

_{P}, which is a function of _{P}(_{P}(_{Threshold}. Then both the threshold current _{th} and the threshold charge, _{th} = _{th} × _{P} reaches _{Threshold}.

Derivation of the Strength–duration relationship. _{th}) decreases as the _{th}) and

Then the critical condition is:

_{th} and _{th} can be written as functions of _{Threshold}:

Since _{P}(_{th}, _{th} and _{th}, which are calculated with a set of modeling parameter [_{rh}. This is because the _{P} will saturate at a maximum value, _{Pmax}, when _{Pmax}, as shown in

Meanwhile,

Since _{rh} is a constant, _{th} increases linearly with _{Pmax}, as shown in

The physical meaning of _{rh} is the threshold current when _{Pmax} = _{Threshold}. Meanwhile, the non-linear curve of _{th} vs. _{rh} and linear curve of _{th} vs.

Rather than infinitely approaching to the _{rh} as the case in Weiss's strength–duration equation, the threshold current curve will be equal to the _{rh} when _{Pmax}.

Rather than being a completely straight line, the threshold charge curve is linear only when _{Pmax}. When the

These two major special differences with the Weiss's equation have already be confirmed by previous research (Friedli and Meyer,

Moreover, it also explains why such relationship can only be applied for negative monophasic square current waveform. Because the voltage waveforms differs with the current waveforms, inducing a more complicated trend without a stable _{rh}, which was observed in other researches (Friedli and Meyer,

_{th}) and the _{th}) and

Then, we will derive the LNP model. The LNP model is a simplified functional model of neural spike responses (Schwartz et al.,

Actually, the Poisson distribution and exponential distribution describe the same stochastic process. If the Poisson distribution provides an appropriate description of the number of the occurrences per interval of time, then the exponential distribution will provide a description of the time interval between occurrences.

The Poisson distribution is as follow:

The exponential distribution is as follow:

These two distributions share the same λ. Apparently, in the C-P theory, if the generation of action potential can be described by exponential distribution, it surely can be described by Poisson distribution.

As follow is the derivation of LNP model.

The white noise involved in LNP model can be simplified as a triangle wave series of frequency _{w} as shown in

Derivation of LNP model from C-P theory. _{e} curve; _{e}curves of the noise waveforms in _{e} vs. the noise amplitude _{w}.

Only part of the voltage can exceed the _{Threshold}. As explained in _{λ} of the λ curve within a period _{e} for Poisson distribution can be calculated based on the _{λ}:

which is the blue straight line in _{e} curve are of the same area, so they will induce the same statistical results.

So the probability calculus equation can be rewritten as:

The corresponding Poisson distribution is:

By increasing the noise amplitude _{w}, _{λ} will also increase, result in an increasing _{e} as shown in _{λ} is a function of _{w}, and λ_{e} is a function of _{λ}, λ_{e} is also a function of _{w}, shown as the non-linear curve in _{w} induces a non-linear increment of λ_{e} happened in LNP model. Because the expression of _{λ} is a piecewise function of _{w}, the exact function λ_{e}(_{w}) can only be calculated numerically with a fixed α, β, _{Threshold}, and _{e}(_{w}) is not available.

In summary, we propose a new theory, named the Circuit-Probability theory, to unveil the “secret” of electrical nerve stimulation, essentially explain the non-linear and resonant phenomena observed when nerves are electrically stimulated. In this theory, an inductor is involved in the neural circuit model for the explanation of frequency dependent response. Furthermore, predicted response to varied stimulation strength is calculated stochastically. Two empirical models, strength-duration relationship and LNP model, can be theoretically derived from C-P theory. This theory is shown to explain the complex non-linear interactions in electrical nerve stimulation and fit

All datasets generated for this study are included in the article/

This animal study was reviewed and approved by IACUC of National University of Singapore.

The theory was developed by HW, JW, and TH. The modeling work was carried out by HW. The major framework of experiment design was carried out by HW, JW, and XT. The experiments of CP nerve stimulation were carried out by SL, JW, XT, and HW. The experiments of TA muscle stimulation were carried out by JW, XT, and HW. The experiments of cortical stimulation were carried out by XT, JW, and HW. The experiments of pelvic stimulation were carried out WP, JW, and HW. The experiments of stimulus artifact recording with high sampling frequency system were carried out by KN, XT, JW, and HW. The data analysis was carried out by HW, JW, XT, and WP. The manuscript was written by HW, JW, and XT. NT and CL provided general guidance of the project. All authors discussed the experimental results and contributed to the final version of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank for the experiment setup support from Han Wu, Shih Chiang Liu, Astrid, Shuhao Lu, Li Jing Ong, and Dian Sheng Wong. We also would like to thank for the animal experiment support from Gammad Gil Gerald Lasam. We have our special acknowledgment to James T. Fulton for his pioneer research of neuroscience published on the Internet.

The Supplementary Material for this article can be found online at: