Multi-Tongue Frequency Fractal Dynamics in Hodgkin-Huxley Neurons Induced by Temporal Interference Stimulation

Abstract

We investigate neuronal excitability in the Hodgkin-Huxley model under temporal interference (TI) stimulation in a previously unexplored sub-Hz resonant regime and uncover a striking nonlinear response that we term 'multi-tongue frequency fractals'. Unlike single-frequency driving, which yields a smooth resonant valley, dual-frequency excitation fragments this response into a hierarchy of sharply modulated tongues whose number and structure grow with observation time, revealing clear self-similar architecture. These features emerge from transitions between non-cascaded and cascaded high-harmonic and sub-harmonic generation as detuning varies, and are maximized near the intrinsic ionic timescale at omega ~ 0.2 rad/s. Parameter sweeps show that the fractal count is higher for higher potassium conductances, lower sodium conductances and lower leak conductances. These results demonstrate that TI stimulation can elicit rich, hierarchically organized frequency responses even in classical excitable membranes, revealing fractal organization in Hodgkin-Huxley dynamics.

Figures (5)

• Figure 1: Stimulation threshold amplitude $J_y^*$ under (a) single-frequency and (b) dual-frequency excitation. In (a), $J_y^*$ reaches its minimum near $\omega \approx 0.2~\text{rad/s}$. In (b), the dual-frequency drive produces a broader response exhibiting multiple minima of $J_y^*$ corresponding to "multitongue fractals".
• Figure 2: (a) Stimulation threshold amplitude $J_y^*$ under dual-frequency excitation for different maximum integration times $T_{\max}$, showing the emergence of tongues within valleys like fractals as the observation window increases. As $T_{\max}$ increases, additional tongue-like substructures appear within the primary valleys, producing a self-similar, fractal-like frequency response. Panels (b)-(d) illustrate three representative contrasting dynamical behaviors responsible for this structure: (b) non-cascaded high-harmonic excitation at $\Delta\omega = 0$, (c) cascaded high-harmonic excitation at $\Delta\omega \approx 0.094~\text{rad/s}$, and (d) cascaded subharmonic excitation at $\Delta\omega \approx 0.138~\text{rad/s}$. The coexistence and transition among these distinct response regimes collectively generate the hierarchical “frequency fractals” observed in (a).
• Figure 3: Stimulation threshold amplitude $J_y^*$ under dual-frequency excitation for different primary drive frequencies $\omega$ as a function of detuning $\Delta\omega$. A fractal-like modulation of the response is most prominently observed near the resonant condition at $\omega = 0.2~\text{rad/s}$, where the system displays the widest spread and highest complexity of oscillatory structures. As $\omega$ increases away from resonance, these frequency-fractal features become confined to a progressively narrower detuning range and their intensity is significantly reduced, leading to smoother response curves.
• Figure 4: (a-c) Effect of Hodgkin--Huxley model parameters viz. $g_{\mathrm{Na}}$, $g_{\mathrm{K}}$ and $g_{\mathrm{L}}$ on the dual-frequency response respectively. Although higher $g_{\mathrm{Na}}$, lower $g_{\mathrm{K}}$ and higher $g_{\mathrm{L}}$ promote excitability, they do not favor the formation of multitongue frequency fractals.
• Figure 5: Fractal count maps showing how the sodium conductance $g_{\mathrm{Na}}$ and potassium conductance $g_{\mathrm{K}}$ together determine the strength of multifractal behaviour for different values of leak conductance $g_{\mathrm{L}}$. Panels (a)--(e) correspond to increasing $g_{\mathrm{L}}$ from $0.1$ to $0.5~\mathrm{mS/cm^2}$. For low $g_{\mathrm{L}}$, a large region of the $(g_{\mathrm{Na}}, g_{\mathrm{K}})$ space supports high fractal counts, indicating strong nonlinear interactions between sodium excitation and potassium recovery currents. As $g_{\mathrm{L}}$ increases, this high-fractal region gradually shrinks and shifts toward higher $g_{\mathrm{K}}$ and lower $g_{\mathrm{Na}}$, and most of the parameter space eventually becomes non-fractal (dark-blue region).