Abstract
We introduce a novel approach of stabilizing the dynamics of excitation waves by spatially extended subthreshold periodic forcing. Entrainment of unstable primary waves has been studied numerically for different amplitudes and frequencies of additional subthreshold stimuli. We determined entrainment regimes under which excitation blocks were transformed into consistent 1:1 responses. These responses were spatially homogeneous and synchronized in the entire excitable medium. Compared to primary pulses, pulses entrained by secondary stimulations were stable at considerably shorter periods which decreased at higher amplitudes and greater number of secondary stimuli. Our results suggest a practical methodology for stabilization of excitation in reactiondiffusion media such as nerve tissue with regions of reduced excitability.
Background
Dynamics of excitation waves in reactiondiffusion media can be altered by spatiotemporal periodic forcing. Additional (secondary) periodic stimulations superimposed on primary forcing may alter the primary excitation waves and entrain (lock) them to the period of secondary stimuli. Locking of primary waves to the period of secondary stimulations occurs at particular values of forcing periods and amplitudes. This resonant shift is characterized by Arnold tongues which determine the margins of different types of M : N (M ≥ N, N ≥1) locking responses as a function of amplitude and frequency of external forcing [1,2].
It was found that the phenomenon of locking manifests itself in different ways depending on the spatiotemporal complexity of the particular reactiondiffusion system. For example, under the periodic forcing, spatially uniform twodimensional BZ reaction oscillations were transformed into standing wave type labyrinths of complex geometry [1]. It was also demonstrated that onedimensional Turing patterns can be modulated by spatiotemporal forcing in the form of a travelling wave [3].
A similar type of resonant behaviour of Turing patterns was observed in experiments with photosensitive chemical reactions [4]. It was also shown that the presence of large amplitude periodic forcing in a onedimensional bistable reactiondiffusion medium resulted in bifurcating of an originally stable wavefront into two counter propagating wavefronts [5]. In contrast, modulating the intensity and frequency of the periodic forcing controlled the trajectory and rotational frequency of twodimensional spiral waves in the Oregonator model [6].
Additional periodic forcing can be also applied for locking of primary waves in some biological excitable media. A typical example of practical realization of such resonant dynamics is a posttraumatic adjustment of excitation in nerves with impaired excitability. It was shown that impaired excitation may be restored by applying additional functional electrical stimulation using implantable [7] or body surface stimulation electrodes [810]. This method has been confirmed as an effective tool for restoration of movement of paralyzed muscles in individuals with a variety of neurological impairments [11].
Usually, after severe neuromuscular injuries nerve conductivity is significantly reduced, which in turn, prevents the passage of excitation waves through neuromuscular transmitters. Under these circumstances propagation of excitation pulses is marginally stable and implementation of functional electrical stimulation necessitates a significant increase of frequencies and amplitudes of additional electrical stimuli. The latest, instead of stabilization of propagation, can facilitate conduction blocks and may completely disrupt the process of training paralyzed muscles.
To address this deficiency we investigate a new approach of entrainment of marginally stable excitation waves by spatially extended low amplitude subthreshold forcing. We demonstrate that such subthreshold forcing can transform excitation blocks to stable 1:1 responses synchronized in the entire excitable medium.
Methods
Dynamics of conduction in nerves is simulated by FitzhughNagumo type reactiondiffusion equations with just two fundamental excitation and postexcitation recovery variables [12,13]. Unlike our recent work where we studied the dynamics initiated by a single excitation source [14], the adjusted model is set up to reflect the interference of several sources which deliver multiple stimuli of different amplitudes. In particular, the main source delivered localized overthreshold stimuli applied at the beginning of the cable and a set of additional sources originated secondary subthreshold forcing which was extended throughout the entire cable (Eqs. 1).
Here u and v are dimensionless excitation and recovery functions, respectively. ε is a small parameter and v_{r }is the excitation threshold. λ and ζ control the rates of changes of excitation and recovery functions, respectively.
The system of Eqs. 1 was solved numerically in a onedimensional cable of finite length using a second order explicit difference scheme with zero flux boundary conditions [15]. Spatial, Δx, and temporal, Δt, steps used in the numerical integration were equal to 0.23 and 0.0072, respectively. The cable length, L, was equal to 150Δx. Primary forcing, P (x, t), was a train of rectangular pulses with duration 100Δt, an overthreshold amplitude A_{0 }and period T_{0}. Primary stimuli were applied near the left end of the cable between x = 2Δx and x = 15Δx. Secondary forcing {S (x  x_{i}, t), i = 1,2,..., n} also a rectangular pulse train with duration 100 Δt, was delivered at n equidistant locations between x = 40Δx and x = 140Δx. The secondary stimuli were simultaneously activated after the wavefront which resulted from the first primary stimulation arrived at the end of the cable. The amplitude and period of secondary stimulations were equal to A and T, respectively.
Unless mentioned otherwise, values of parameters in all computations were ε = 0.1, λ = 0.4, ζ = 1.2, A_{0 }= 1.4, α = 0.31, β = 0.0025 and n = 6. As in [14] we used a simplified primary rate dependent excitation threshold given by linear equation v_{r }= α  βT_{0}, α > 0, β > 0. The duration of a pulse, T_{h}, was measured as the time interval between consecutive intersections of u and v near their rest and excited states, respectively (Figure 1). The steady state value of T_{h }was computed after 80 primary stimulation periods.
Figure 1. Excitation, u, (solid line) and recovery, v, (dashed line) variables at x = L /2 for the short steady state pulse. Left and right intersections between u and v mark the beginning and the end of the pulse, respectively.
Results
In the absence of secondary stimulation (A = 0), at long periods T_{0 }the variable v has enough time to reach its steady state v_{r }before the next stimulus is applied. However, as T_{0 }is reduced to a critical limit, T_{end}, the system does not respond to every stimulus and exhibits unstable M : N (M > N ) excitation blocks which occur due to incomplete recovery of control variable v. Figure 2 compares phase portraits of the system at two values of T_{0}. At T_{0 }= 60, u closely follows its nullcline and v almost completely recovers to its steady state value of 0.16 as depicted by the intersection of the u  v nullclines. However, at T_{0 }= T_{end }= 30, deviations of u and v from their nullclines are quite significant, thereby v recovers to a value which is much higher than the corresponding threshold of 0.23.
Figure 2. Phase portraits of the system at x = L/2 for T_{0 }= T_{end }= 30 (dashed contour) and T_{0 }= 60 (solid contour). Nullcline of u is given by thick Nshaped line. Dashed and solid lines with intercepts at v_{r }= 0.23 and v_{r }= 0.16 are nullclines of v for T_{0 }= 30 and T_{0 }= 60, respectively.
Analysis of pulse duration T_{h }at the critical primary period T_{end }= 30, as a function of secondary frequency F = T ^{1 }and forcing amplitude A ≥ 0 revealed a variety of entrainment regimes (Figure 3). We found that the system did not respond to secondary stimuli at amplitudes which were smaller than critical values depicted by the curve with circular markers. For the amplitudes above this curve and frequencies smaller than F_{0 }we observed intermediate M : M responses with M greater than one.
Figure 3. Locking margins for different frequencies and amplitudes of secondary stimuli for T_{0 }= 30, A_{0 }= 1.4, .
For even greater amplitudes, above the upper curve with square markers, the system locked to secondary stimuli with consistent 1:1 responses. It should be noticed that such locking occurred over a wide range of secondary frequencies, and that secondary amplitudes were five times smaller than the amplitudes of primary stimulations. We also observed that for frequencies greater than F_{0 }the entrainment of blocked excitation occurred without intermediate M:M responses (Figure 3).
Spatiotemporal contours of u shown in Figure 4 demonstrate expected unstable responses to primary stimulation at T_{0 }< T_{end}. Temporal dynamics of u and v, as well as spatiotemporal contours of u, show 3:2 excitation blocks (Figure 4,5). However, in the presence of secondary stimulations such unstable responses can be entrained and stabilized by secondary driving even at T_{0 }< T_{end}. Indeed, Figure 6 demonstrates that 3:2 blocks can be transformed into stable 1:1 responses which evolve homogeneously in the entire cable except for short segments located near the site of primary stimulation.
Figure 4. Grayscale patterns in the spatiotemporal evolution of u. Corresponding scale bar is shown on the right.
Figure 5. Dynamics of u (solid line) and v (dashed line) for three consecutive cycles at T_{0 }= 28, A_{0 }= 0, .
Figure 6. Grayscale bands depicting synchronization of u for T_{0 }= T = 28 and A = 0.1A_{0}. Corresponding scale bar is shown on the right.
Formation of these fully synchronized responses are preceded by very short (~ 0.02T_{0}) transient periods during which standing wave type oscillations of u rapidly saturate at constant excitation levels (Figure 7).
Figure 7. Temporal evolution of u near the bottom of the second band as depicted in Fig. 6. Profiles are shown for three equidistant moments of time, starting at t_{init }= 136, δt = 0.36.
When compared with primary forcing alone (A = 0), secondary subthreshold stimuli facilitate development of stable pulses shorter than those at T_{end }= 30. Figure 8 shows that secondary stimuli of higher amplitudes sustain shorter entrained pulses at progressively smaller stimulation periods. The trend can be slightly augmented for higher coefficients β when T_{end }decreases as shown in Figure 9.
Figure 8. Dependence of T_{h }on T shown for different amplitudes of secondary stimuli at T_{0 }= T. Ends of the curves at T_{end }for different amplitude ratios are indicated by corresponding markers.
Figure 9. Dependence of T_{end }on for two values of β at .
Stabilization of the system due to secondary driving can also be achieved using a greater number of secondary stimulation sources. Correspondingly, Figure 10 demonstrates that an initial twofold increase of the number of secondary sources from 2 to 4 extends the region of stability towards shorter values of T_{end}. Further increase of the number of sources saturates these changes at progressively shorter values of T_{end }for smaller coefficients β.
Figure 10. Dependencies of T_{end }on n for different β, α = 0.67 and .
Conclusions
In summary, we demonstrated that additional subthreshold driving stimuli can entrain otherwise unstable primary reactiondiffusion waves and transform M : N excitation blocks into stable 1:1 spatially homogeneous responses synchronized in the entire cable. Compared to pulses resulting from primary forcing alone, pulses entrained by secondary stimulations were stable at considerably shorter periods. These periods decreased at higher amplitudes and greater number of secondary stimuli. We also found that locking to stable 1:1 responses occurred over a wide range of secondary frequencies. In addition, the subthreshold secondary amplitudes were a factor of five smaller than the amplitudes of primary stimuli. Our results outline the possibility of entrainment of reactiondiffusion waves by subthreshold additional driving and may be applied for stabilization of excitation in nerves with regions of impaired excitability [16].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JS conceived, developed and designed the theory and numerical experiments. VV performed the numerical simulations and has been involved in drafting and revising the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
We would like to thank Vladimir Polotski and Shyam Aravamudhan for useful discussions and continuous interest to our work. We also thank Alan Covell for editorial comments.
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