The Mathematics of Stress and Stability
The hidden connection between brain cells and stress hormones reveals a universal language of biological rhythms.
Imagine your nervous system as a sophisticated orchestra, where each neuron plays its part in perfect timing to create the symphony of your thoughts, emotions, and physical responses. Similarly, your endocrine system functions as an elaborate communication network, releasing hormones that travel through your bloodstream to coordinate everything from stress responses to metabolism. What if these two systems spoke the same mathematical language?
This is the story of the FitzHugh-Nagumo model—a brilliant simplification of neural activity that has unexpectedly found applications far beyond its original purpose, helping us decipher the complex rhythms of the hypothalamic-pituitary-adrenal (HPA) axis, your body's central stress response system. The connection reveals fundamental principles that govern excitation and stability across biological systems, offering insights into conditions from depression to chronic fatigue.
The same mathematical principles that describe neural firing also govern hormonal pulsatility, revealing deep biological connections between seemingly different systems.
In 1961, scientists Richard FitzHugh and J. Nagumo sought to simplify the notoriously complex Hodgkin-Huxley model of neural activity1 . Their goal was to capture the essential features of neural excitation without the mathematical complexity that made the original model challenging to work with.
The resulting FitzHugh-Nagumo model consists of just two elegant equations that mimic the electrical behavior of excitable cells1 :
dV/dt = V - V³/3 - W + I
dW/dt = ε(V + a - bW)
Here, the parameter I represents external stimulation to the neuron, while a, b, and ε are controlling parameters that influence the system's dynamics. The beauty of this model lies in its ability to capture the core behavior of neuronal excitation with minimal mathematical complexity.
What makes the FitzHugh-Nagumo model so valuable is its accurate portrayal of threshold behavior in neurons3 . A small stimulus simply decays back to equilibrium, while a stimulus exceeding a critical threshold triggers a full action potential—an all-or-nothing response fundamental to neural signaling.
The voltage variable increases quickly when stimulated
The restorative variable catches up, bringing the system back toward equilibrium
A brief period where the neuron is less responsive to additional stimulation
When extended to neural networks using reaction-diffusion equations, the model can simulate wave propagation of electrical activity, mimicking how signals travel along neurons and neural pathways.
| Variable | Biological Correspondence | Mathematical Role |
|---|---|---|
| V | Neuron membrane potential | Primary excitable variable |
| W | Recovery processes (potassium channel activation, sodium channel inactivation) | Restorative variable |
| I | External stimulus to neuron | Input parameter |
| ε | Timescale separation between fast V and slow W | Rate parameter |
| a, b | Intrinsic membrane properties | Shape parameters |
While the FitzHugh-Nagumo model originated in neuroscience, its applications extend to neuroendocrinology—particularly in understanding the hypothalamic-pituitary-adrenal (HPA) axis. This sophisticated system comprises2 4 :
The HPA axis operates through a precise cascade of hormonal signals: CRH → ACTH → cortisol4 . Cortisol then completes a negative feedback loop by inhibiting both the hypothalamus and pituitary, maintaining system balance4 .
Unlike a simple on-off switch, the HPA axis exhibits complex ultradian rhythms—pulsatile releases of hormones occurring approximately 15-22 times over 24 hours5 . These pulses are superimposed on a circadian pattern, with cortisol levels typically peaking 30-45 minutes after waking, gradually declining through the day, rising slightly in late afternoon, and reaching a trough around midnight4 .
This rhythmic activity ensures that stress responses are appropriately timed and proportional. Disruptions to these patterns are associated with various conditions, including chronic fatigue syndrome, insomnia, and burnout4 .
To understand how mathematical neuroscience intersects with neuroendocrinology, consider a pivotal study that applied mathematical modeling to decipher the HPA axis's pulsatile dynamics5 . Researchers faced a significant challenge: while they could measure ACTH and cortisol levels, the underlying pulsatile events driving these hormones remained hidden.
The research team developed a novel framework using compressed sensing—a technique for reconstructing sparse signals from limited measurements5 . This approach was particularly suited to hormonal pulses because they represent brief, intense secretory events against a relatively quiet background.
Using compressed sensing algorithms, researchers could reconstruct hidden pulsatile events from hormone concentration data, revealing the precise timing and amplitude of HPA axis activity.
The study enrolled 10 healthy women who underwent rigorous monitoring under controlled conditions5 . The experimental protocol included:
Collection of serum samples every 10 minutes for 24 hours via indwelling intravenous catheter
ACTH levels measured using the Nichols Allegro HS-ACTH kit, cortisol measured via chemiluminescence assay
Participants followed standardized sleep-wake cycles and meal times to minimize external influences
The mathematical model treated hormonal secretion as an impulse train—a series of brief, intense secretory events—and incorporated first-order kinetics for hormone clearance5 . The team then employed a coordinate descent approach to simultaneously recover both the model parameters and the timing/amplitudes of secretory events.
| Age | BMI (kg/m²) | Exclusion Criteria |
|---|---|---|
| 23-44 | 20.7-29.9 | Current medical problems, glucocorticoid use within past year, estrogen/progesterone use within past 4 months, diagnosis of depression |
The analysis successfully recovered 15-18 pulsatile events over 24 hours, aligning with physiological expectations5 . More importantly, the model quantified not just the timing but also the amplitudes of these hidden secretory events, providing unprecedented insight into HPA axis dynamics.
Determining whether cortisol dysregulation originates in the hypothalamus, pituitary, or adrenal glands
Informing timing and dosage of cortisol replacement therapies
Understanding how various disorders disrupt normal HPA axis function
| Research Tool | Function | Application Example |
|---|---|---|
| Nichols Allegro HS-ACTH Kit | Measures ACTH concentration in plasma | Quantifying ACTH levels in serum samples |
| Chemiluminescence Assay (Beckman Coulter) | Determines cortisol concentration | Measuring serum cortisol levels |
| Intravenous Catheter | Enables frequent blood sampling | Collecting blood every 10 minutes over 24 hours |
| Compressed Sensing Algorithms | Reconstructs sparse pulsatile signals | Determining timing and amplitude of hormonal secretory events |
| Constant Routine Protocol | Standardizes environmental influences | Minimizing effects of stress, posture, eating on hormone levels |
The journey from the FitzHugh-Nagumo model to HPA axis dynamics reveals a profound truth about biological systems: seemingly disparate processes often share common mathematical principles. The same dynamics that govern a neuron's electrical spike appear in the pulsatile release of stress hormones, suggesting deep evolutionary conservation of excitation-recovery mechanisms.
This interdisciplinary approach offers promising avenues for understanding and treating conditions characterized by HPA axis dysregulation, including depression, anxiety disorders, and post-traumatic stress disorder6 . By speaking the common language of mathematics, researchers can decode the complex conversations between our neurons and hormones, potentially developing more targeted interventions for when these systems fall out of balance.
As research continues, we may discover that these mathematical patterns extend even further—perhaps representing universal principles governing excitation, recovery, and rhythm across the biological world.