Ion gradient-driven bifurcations of a multi-scale neuronal model
Metabolic limitations within the brain frequently arise in the context of aging and disease. As the largest consumers of energy within the brain, ion pumps that maintain the neuronal membrane potential are the most affected when energy supply becomes limited. To characterize the effects of such limitations, we analyze the ion gradients present in a conductance-based (Morris–Lecar) neural mass model. We show the existence and locations of Neimark–Sacker and period-doubling bifurcations in the sodium, calcium, and potassium reversal potentials and demonstrate that these bifurcations form physiologically relevant bounds of ion gradient variability. Within these bounds, we show how depolarization of the gradients causes decreased neural activity. We also show that the depolarization of ion gradients decreases inter-regional coherence, causing a shift in the critical point at which the coupling occurs and thereby inducing loss of synchrony between regions. In this way, we show that the Larter-Breakspear model captures ion gradient variability present at the microscale level and propagates these changes to the macroscale effects such as those observed in human neuroimaging studies.
Figure: Flip bifurcation in the potassium ion gradient lengthens refractory period. A. Family of local limit cycle continuations originating at the potassium gradient Hopf point (𝑉K ≈ −1.102). Unlike the sodium and calcium gradients, there is no torus (Neimark–Sacker) bifurcation between the Hopf point and the flip (period-doubling) bifurcation, which here occurs at 𝑉K ≈ −0.61 (shown in red and labeled PD). B. Mean excitatory voltage 𝑉 waveforms under three conditions: baseline reversal potential (𝑉K = −0.7), depolarized just over the torus bifurcation (𝑉K = −0.59), and significantly hyperpolarized (𝑉K = −1.0). C. Attractors plotted in 𝑊 − 𝑍 phase space at each of the three 𝑉K values listed in B (see legend in B for colors). Crossing the flip bifurcation into the physiologically relevant space below (yellow to blue) only shrinks the refractory period and induces more chaotic activity (increased filling of phase space by the trajectories). Moving further from the flip bifurcation (i.e., the gradient is more hyperpolarized, shown as blue to orange) increases the chaotic activity, but because there is no torus bifurcation the orbits do not collapse to a basin as seen in sodium and calcium. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
For more Information see:
Chesebro A, Mujica-Parodi L, Weistuch C. Ion gradient-driven bifurcations of a multi-scale neuronal model. Chaos, Solitons & Fractals. 167, 113120 (2023); doi: 10.1016/j.chaos.2023.113120. PDF
