Mathematical modeling plays a central role in uncovering and quantifying the physiological mechanisms underlying neuronal systems, providing a principled framework for understanding brain function and optimizing therapeutic interventions such as neuromodulation. Neuromodulation employs external stimuli—such as electrical pulses or focused ultrasound—to regulate neural activity and treat neurological disorders including Parkinson’s disease and depression.

Neuroscience spans multiple scales, from single-neuron dynamics (e.g., membrane potential evolution) to population-level descriptions (e.g., mean firing rates). Correspondingly, mathematical models can be studied through both numerical simulation and analytical approaches. While simulation remains a fundamental tool in computational neuroscience, analytical solutions—when available—offer complementary advantages: they enable efficient model implementation and provide mechanistic insight by revealing explicit relationships between key variables.

This presentation highlights two recent advances:

(1) We derive an analytical expression for the firing rate of the leaky integrate-and-fire (LIF) neuron driven by inputs with arbitrary (non-Gaussian) distributions, bridging the gap between idealized assumptions and experimentally observed physiological variability. This result enhances both the interpretability and computational efficiency of LIF-based modeling at single-neuron and network levels.

(2) We develop a mean-field model of the thalamic ventral intermediate nucleus (Vim) under deep brain stimulation (DBS), a widely used therapy for movement disorders. Our analysis reveals a mechanistic explanation for the efficacy of high-frequency (≥100 Hz) DBS: it preferentially recruits inhibitory neuronal populations, thereby stabilizing network activity.

Together, these results demonstrate how analytical and mean-field modeling approaches can provide mechanistic insight into neuromodulation and guide the design of more effective therapeutic strategies.