1 Introduction

Neurons are elementary units of the nervous system and have the ability to conduct signals rapidly over large distances. They receive, process, and transmit information by generating characteristic electrical pulses called action potentials or voltage spikes that occur when the membrane potential rapidly rises and falls. With the presence of external sensory stimuli, including light, sound, taste, smell, and touch, neurons change their activities by firing sequences of spikes in various temporal patterns. Although spikes can vary in duration, amplitude, and shape, they are typically treated as identical stereotyped events in neural coding studies. A number of mathematical models have been proposed for describing the phenomena of spikes, such as the Hodgkin-Huxley (HH) model ^[1], Morris-Lecar (ML) model ^[2], FitzHugh-Nagumo (FHN) model ^[3].

The processes that generate only a single spike correspond to the depolarization, repolarization, and refractory period of the neuron. These processes are relatively simple. Repetitive spiking usually leads to more complicated dynamics, especially the bursting phenomenon. Bursting (burst firing) in a neuron is the potential, or chemical concentration changes between repetitive spiking and a quiescence state ^[4-5], and it has been found in many types of neurons, such as thalamic neurons ^[6] and dopamine-containing neurons of the mammalian midbrain ^[7]. Bursting patterns named Types Ⅰ, Ⅱ, ..., Ⅴ have been introduced after Rinzel's work on the classification of bursting ^{[4, 8-10]}. In view of a geometric bifurcation theory, Izhikevich ^[5] presented a complete classification, and suggested that the patterns of bursters should be named on the basis of the names of the two bifurcations involved instead of descriptions or the type plus the number, such as square-wave (Type Ⅰ) burster, parabolic (Type Ⅱ) burster, and elliptic (Type Ⅲ) burster.

Apart from intrinsic properties of neurons, oscillatory activities may result from neural network properties, such as coupling strength and time delay ^[11]. Synchronization of coupled neurons is a phenomenon that has been studied extensively in Refs. [12]-[14]. Synchronization can be obtained for weakly connected networks ^[12], but not all weakly connected neurons can be synchronized ^{[5, 15-16]}. While the coupling of neurons can be considered as a passive control strategy that has been a research topic of great interest over the past few decades, external stimuli can be regarded as active control strategies and have been used in some applications. Taking voluntary movement of the human body as an example, one can control some neurons to make coupled neurons do something specific, where the motor information is calculated in cortical areas and carried by upper motor neurons along the spinal cord to activate the lower motor neurons, and in turn to make the muscles contract. Motor cortex neurons were used to record signals for real-time device control in rats ^[17] and monkeys ^[18]. The study of the control (stimulus) mechanism among neurons has the potential to provide more accurate and effective stimuli in therapies by electrical nerve stimulation, such as transcutaneous electrical nerve stimulation (TENS) and neuromuscular electrical stimulation (NMES). Neuropeptide release by the electrical stimulation with different frequencies was introduced in Ref. [19]. In addition, a washout filter-aided dynamic feedback control was introduced to the Hindmarsh-Rose model neuron to change its type from Type Ⅱ to Type Ⅰ ^[20]. In Ref. [21], a feedback control was proposed for synchronization of two coupled FHN models. In Ref. [22], a kind of neural network was synchronized by using a single controller.

Some biological phenomena of neurons can be explained from the viewpoint of firing activity and synchronization, but the control mechanism of neurons has not been fully understood yet. It is still unclear what controllers are actually used in real interactions of neurons, and how to control their activities precisely. In the control design of coupled neurons, uncertainties should be taken into account, because it is hard to have the model parameters precisely measured, and the coupled neural systems are usually subject to the external disturbance. Thus, it is more reasonable to design the controller on the basis of measured signals. In this study, the track control of an uncertain neuron by stimulating another coupled uncertain neuron is studied. The following considerations distinguish this study from the previous works. (ⅰ) Uncertainties in neurons are taken into account, and only a controller can be used to make another neuron track a specified signal. (ⅱ) Compared with the pinning control in neural networks, the strength of coupling has no particular limit, provided that it is not equal to 0. (ⅲ) A robust controller with saturation function is presented in this paper.

2 Statement of the problem

The simplest control process between two neurons, as shown in Fig. 1, is considered. An appropriate stimulus on the neuron 1 is designed to make the coupled neuron 2 generate a desired action potential. The problem is a kind of track control of a two-dimensional system with one controller, which can be mathematically described as

(1)

where u₁ and u₂ denote the membrane potentials of the neuron 1 and the neuron 2, respectively. In the HH model ^[1], w₁, w₂∈ℝ³ denote the three types of ion channels, i.e., a sodium channel with the index Na, a potassium channel with the index K, and an unspecific leakage channel with the resistance R. In the ML model ^[2], w₁, w₂∈ℝ² (or ℝ¹) denote the Ca and K ion channels (or the K ion channel). In the FHN model ^[3], w₁, w₂∈ℝ¹ denote recovery variables. The system (1) is also a generalized model of some other neuron models, such as the calcium-induced calcium release (CICR) model ^[23]. In mathematics, the functions f₁, f₂, g₁, and g₂ should be continuous and smooth with respect to u₁, w₁ or u₂, w₂. k denotes the coupling intensity, and d_i=d_i(u_i, w_i, t) (i=1, 2) denote the unknown stimuli from the external environment or other adjacent neurons or modeling error.

Assumption 1  The unknown terms d_i vary slowly, that is, there exist small positive numbers ξ_i>0 such that .

The goal is to design an appropriate stimulus I(t) to make u₂ track a desired membrane potential u_d(t) asymptotically and rapidly, that is, u₂-u_d(t)→ 0 as t→+∞ asymptotically and rapidly. Owing to the existence of uncertainties, it may be impossible to achieve such a goal. Therefore, a more realistic goal is to make u₂-u_d(t)→ 0 (t→+∞) asymptotically, as far as possible.

3 Design of the stimulus I(t) 3.1 Observers and the differentiator

For the slowly varied uncertainty d₁, the observer is designed as

(2)

and for the slowly varied uncertainty d₂, the observer can be presented as

(3)

where and are the observed values of d₁ and d₂, respectively, and the parameters k₁, k₂>0 are to be designed.

The principles of designing the observers (2) and (3) are similar to those found in other papers ^[24-25]. In most of these papers, the boundaries of for i=1, 2 were calculated by means of the Lyapunov function. In the present study, the boundaries are estimated using the general solution formulae of ordinary differential equations with constant coefficients.

Theorem 1  Under Assumption 1, consider the observers (2) and (3) for the system (1). , there exist two sufficiently large numbers T_i>0 (i=1, 2) such that

(4)

whenever t>T_i (i=1, 2).

Proof  Differentiating and in observers (2) and (3) with respect to time t and substituting the system (1) into them by and , respectively, give

(5)

for i=1, 2. It follows that

By using the general solution formulae for ordinary differential equations with constant coefficients, we get

and it goes to

(6)

Note that

Thus, , there exist two sufficiently large numbers T_i>0 (i=1, 2) such that Eq. (4) holds. This completes the proof.

According to Eq. (5), the designs of the observers (2) and (3) are similar to those of the first-order low-pass filters. The boundaries in Theorem 1 are also the maximum errors of the first-order low-pass filters.

Moreover, from Eq. (5), one has

Then, it follows that

Thus, one has

(7)

Hence, is bounded for the system (1) and observers (2) and (3) under Assumption 1. Similarly, one can prove that the jth derivative is also bounded for j=1, 2, ..., n. In the following, the estimated value of , denoted by v₂, is introduced by a traditional differentiator,

(8)

where γ is a small positive number. The state equation corresponding to the transfer function (8) is

(9)

where p(t) is an intermediate variable. The estimated error is given by the theorem below.

Theorem 2  With the differentiator (8) or (9), the estimated error of has the following estimation. , there exists a sufficiently large T₃>0 such that

whenever t>T₃.

Proof From Eq. (8) or Eq. (9), one has

(10)

which is equivalent to

Again, by using the general solution formulae of ordinary differential equations with constant coefficients, and taking the absolute value on both sides, one has

Note that

(11)

together with Eq. (7), and one obtains

(12)

This theorem holds true, because

3.2 Robust nonlinear control

Let u_d be the known and desired membrane potential for u₂ to track. Then, from the system (1), the error variable u₂-u_d satisfies

(13)

The main idea in the control design is to decompose the right hand side of the first equation of the system (13) into two parts, i.e., the stable term and the rest term. It is required to design an appropriate controller to make the rest term tend to 0 as far as possible, by using the second equation of the system (13). The details are given below.

The first equation of the system (13) can be rewritten as

where a₁>0 is a number to be determined. Let . Then, one has

Due to the existence of the unknown term d₂ in z, only the observed value of is available in the control design. Thus, one has

where . Combining the second equation of the system (13), one has

(14)

Because d₁ and d₂ are uncertain, the precise value of

cannot be obtained, where and g₂(u₂, w₂)= . Note that

Thus, Eq. (14) becomes

(15)

When , where I₀ is to be designed, one has

In this way, the system (1) (or equivalently the system (13)) changes to

(16)

with the following controller:

(17)

where z, v₂, , and are as mentioned above.

Furthermore, I₀ can be designed as follows:

(18)

where μ and β are positive numbers and satisfy and .This is reasonable according to Subsection 3.1. s_α(·) denotes the saturation function, which is defined as

where α>0.

The emphasis should be placed on the saturation function in the proposed controller. In the theory of sliding mode control, the saturation function can be replaced by the sign function, and a better result than that in Theorem 2 can be obtained. However, the sign function generates a chattering phenomenon caused by time delay, small disturbance and so on, when the trajectory is close to the objective trajectory. Although a small chattering phenomenon may not have great effects on the control effect, high-frequency chattering near the objective trajectory has a negative effect on the controller, and the controller will switch at a high frequency with a large amplitude. Thus, it is unreasonable to use the sign function in stimulating a neuron using the high-frequency potential with the large amplitude in practice.

Theorem 3  For the system (16) under the controller defined by Eq. (18), that is, the system (13) (or the system (1)) under the controller defined by Eqs. (17)-(18), it holds that

(Ⅰ) there exists a number T₄≥t₀ such that

(19)

for t≥T₄;

(Ⅱ) for a₁>0 and , there exists a sufficiently large positive number T₅> T₄ such that

(20)

whenever t> T₅.

Proof (Ⅰ) First, it is necessary to prove that there exists T₄≥t₀ such that

(21)

Assume by contradiction that the statement of Eq. (21) is not true. Then, z-u₁>α or z-u₁ < -α for all t≥t₀. It is only necessary to prove the first case because the rest case can be proved with the same method. z-u₁>α holds for all t≥t₀. Then, from the second expression of the system (16), one has

(22)

and . It follows that

(23)

as t→+∞. This is a contradiction for the hypothesis that z-u₁>α holds for all t>t₀. That is to say, the inequality (21) is true.

Secondly, it is necessary to prove that Eq. (19) is true. Assume by contradiction that the statement of Eq. (19) is not true. Then, there exists T'₄> T₄ such that z(T'₄)-u₁(T'₄)>α or z(T'₄)-u₁(T'₄) < -α. For the first case, there exists a T satisfying T₄≤T < T'₄ such that z(T)-u₁(T)=α and z(t)-u₁(t)>α for t∈(T, T'₄). Thus, by the mean value theorem, there exists at least one T'∈(T, T'₄) such that . However, by Eq. (22), one has for all t∈(T, T'₄). This is a contradiction. The second case can also be proved to be contradictory in a similar manner. Thus, Eq. (19) is true.

(Ⅱ) From the proof of (ⅰ), one can conclude that

(ⅰ) if |z(t₀)-u₁(t₀)|>α, then z-u₁≤|z(t₀)-u₁(t₀)| for t∈[t₀, T₄), which can be deduced with the same method as used for Eq. (23) and the result of (Ⅰ);

(ⅱ) if |z(t₀)-u₁(t₀)|≤α, then Eq. (19) holds for t∈[t₀, +∞).

In this proof, we only consider the first case (the case (ⅰ)); the other case can be proved with the same method. According to the general solution formulae, the first expression of the system (16) can be rewritten as

(24)

Note that with Eq. (6), for t> T₄, one has

Thus, Eq. (20) is true, because .

4 Simulation

As an application of the proposed control, the tracking control of the uncertain FHN model with one stimulus, described as

(25)

is studied, where (a, b, c)=(0.7, 0.8, 3), the uncertain terms d₁ and d₂ are d₁=0.1sin() and d₂=-1.2-0.3sin(), and k=0.1. To be more practical, the tracking signal is generated from the following modified ML model:

(26)

where , and (V₁, V₂, V₃, V₄, E(L), E(Ca), g(L), g(K), g(Ca)) =(-0.01, 0.15, 0.1, 0.05, -0.5, 1, 0.5, 2, 1.2). Two classical bursters for the desired action potential are tracked. The two bursters are called the "fold/Hopf" burster (see the blue solid lines in Fig. 2 with E(K)=-400) and "fold/homoclinic" burster (see the blue solid lines in Fig. 3 with E(K)=-0.7) according to the geometric bifurcation theory ^[5]. To classify the bursters, there are two important bifurcations involved, i.e., bifurcation from a quiescent state to repetitive spiking, and bifurcation from a spiking attractor to a quiescent state. A "fold/Hopf (homoclinic)" burster means that a fold bifurcation causes a neuron to change from a quiescent state to repetitive spiking, and a Hopf (homoclinic) bifurcation causes a neuron to change from a spiking attractor to a quiescent state. Usually, the two bifurcations can be analyzed by nullclines of fast subsystems and phase planes ^[26]. Our goal is to make u₂-V(:=u_d)→ 0 as t→+∞, as far as possible.

As shown in Eqs. (5) and (10), the observed values and the differentiated value only have connection with uncertainties d₁, d₂, and , and they have no connection with the state variables in the neuron model. Thus, we only give the case in the "fold/Hopf" burster. In the application of the control strategy proposed in Section 3, the parameter values are fixed, (k₁, k₂, γ, μ, β, α, a₁)=(10, 10, 0.1, 2, 2, 0.5, 10). Figures 2 and 3 show that the "fold/Hopf" burster and "fold/homoclinic" burster generated by the ML model (26) (see the blue solid lines) have been tracked well and quickly by the second neuron in the FHN model (25) (see the red dotted lines). As shown in Figs. 4-6, the errors of estimation of the two observers and the differentiator are very small after a period of time. However, at the start, the error of estimation is relatively large. This is connected with the selection of initial values of the systems (2), (3), and (9).

5 Conclusions

This study investigates the track control of the membrane potential of a neuron by stimulating an adjacent coupled neuron. Owing to the presence of uncertainties from model parameters and external disturbances, estimations of the uncertain terms are presented in terms of observers and a differentiator constructed using measured signals. The observers and differentiator are primarily first-order low-pass filters, and the errors of their estimations depend on the characteristics of the uncertain terms only. In addition, a robust observer-based controller with a saturation function is presented for tracking the desired membrane potential, and it is robust against uncertainties. It is expected that the proposed controller might help in understanding the mechanism of neuronal control with a simple model, and more complex cases are left for future investigation.

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