Biological degeneracy preserves phase diagrams topology in excitable systems

Inside the living cells that power our hearts and minds, there is a constant, rhythmic electrical struggle. These cells, known as excitable systems, do not simply sit still; they generate pulses of electricity that travel through the body, telling the heart to beat and the brain to think. To understand how these pulses work, scientists build mathematical models—simplified digital representations of the cell's inner workings. These models act like maps, showing how the cell behaves under different conditions. By adjusting the knobs on these maps, researchers can see the cell switch from a calm resting state to a steady rhythm, a chaotic burst, or a dangerous irregularity. Understanding these maps is crucial because the same electrical patterns that keep a heart beating can, when they go wrong, cause fatal arrhythmias or seizures. For decades, scientists have relied on two specific types of maps to study these behaviors: one that treats the cell as a continuous flow of time, and another that treats it as a series of discrete steps. While these maps look different on the surface, they have long been suspected of sharing a deep, hidden connection.

A team of researchers from Brazil, led by R. V. Stenzinger and M. H. R. Tragtenberg, has now confirmed that these two distinct maps are far more similar than anyone realized. In their recent work, they explored the parameter spaces of two famous models: the Hindmarsh-Rose model, which describes the cell's voltage as a smooth, flowing curve, and the KTz model, which describes it as a series of jumps. In the standard version of the Hindmarsh-Rose map, scientists usually adjust the "external current"—a constant electrical push applied from outside the cell—to see how the system changes. In the standard KTz map, however, they adjust the "reversal potential," which is the specific electrical voltage at which an ion channel stops pushing or pulling, effectively acting as a natural internal switch. For years, these two maps were studied separately, with the assumption that their different control knobs would lead to different landscapes of behavior. The researchers set out to test a bold idea: what would happen if they swapped these knobs? What if they used the external current to drive the KTz map and the reversal potential to drive the Hindmarsh-Rose map?

To find the answer, the team did not just guess; they used rigorous mathematical stability analysis to draw the boundaries of where the cell would oscillate versus where it would sit still. They treated the slow-moving parts of the cell's machinery as nearly frozen, allowing them to calculate the precise lines where the system would lose its stability and begin to pulse. This calculation revealed a three-dimensional structure in the space of all possible settings, a shape that contains every possible behavior the cell could exhibit. When they sliced through this structure to create new two-dimensional maps, they discovered something remarkable. When they took the KTz model and replaced its internal reversal potential with an external current, the resulting map looked almost identical to the original. The regions of cardiac-like spiking, bursting, and chaotic oscillations appeared in the exact same places. Conversely, when they took the Hindmarsh-Rose model and replaced its external current with a reversal potential, the new map preserved the original layout perfectly. The topological structure—the shape and arrangement of the behaviors—remained unchanged.

This preservation of the map's layout is not a coincidence; it is a consequence of a fundamental property of biology called degeneracy. In complex biological systems, different physical mechanisms can produce the exact same functional result. In the context of these cells, the researchers found that changing the external electrical push has the same mathematical effect as changing the internal voltage threshold where ions switch direction. It is as if the cell has two different ways to achieve the same state of excitement, and the system is robust enough that swapping one for the other does not break the pattern. The researchers demonstrated this by generating a new "excitable" diagram for the KTz model, which now includes the external current, and a new "autonomous" diagram for the Hindmarsh-Rose model, which now relies on the reversal potential with no external push. Both new diagrams displayed the same intricate patterns of behavior found in their original counterparts, including the specific regions where the cell mimics the plateau spikes of heart cells or the subthreshold oscillations of neurons.

The implications of this discovery extend beyond the two models they tested. The researchers showed that this phenomenon is rooted in the basic physics of how cell membranes work. In a real cell, the flow of electricity is determined by the balance between the conductance of ion channels and the voltage difference across the membrane. Mathematically, shifting the conductance or the equilibrium voltage can produce the same effect as adding an external current. This means that the degeneracy observed in the models is likely a universal feature of biological membranes, not just a quirk of a specific equation. It suggests that the brain and heart are built with a redundancy that allows them to maintain their function even when individual components change. If a disease alters the concentration of ions in the blood, shifting the reversal potential, the cell might compensate by adjusting its sensitivity to external signals, or vice versa. This flexibility is what allows biological systems to remain stable in a noisy, changing environment.

The researchers also noted that this finding has direct relevance for medical treatment. If an external electrical stimulus, such as deep brain stimulation, can produce the same effect as a change in ionic balance, then doctors might have more flexibility in how they treat conditions like epilepsy or heart arrhythmias. Conversely, if a drug alters the ionic configuration, it might be possible to counteract it with a specific electrical therapy. The study confirms that the landscape of cellular behavior is far more interconnected than previously thought. By swapping the control knobs on these mathematical maps, the researchers did not scramble the results; they revealed that the underlying structure of life's electrical rhythms is resilient and invariant. The maps of the heart and the mind, it turns out, are drawn with the same ink, regardless of which pen is used to trace them.