The code provided is a simulation of a *Fractional Leaky Integrate-and-Fire Model (FLIF)*, which is an extension of the classical Leaky Integrate-and-Fire (LIF) model. This model is designed to capture certain aspects of neuronal membrane potential dynamics, particularly emphasizing the role of fractional calculus to account for memory effects and anomalous diffusion that are observed in real neurons. ### Biological Basis #### Leaky Integrate-and-Fire Neuron Model The classical LIF model is a simplified representation of a single neuron's membrane potential, wherein: - **Membrane Potential (V):** Represents the difference in electric charge inside and outside the neuron. The potential follows an integrate-and-fire process — integrating synaptic inputs and firing a spike when a threshold is exceeded. - **Leakage (Decay):** The membrane potential experiences natural decay over time towards a resting potential, modeling the passive conduction and leaky nature of neuronal membranes. - **Firing Threshold (vth):** The potential above which the neuron fires an action potential (spike). - **Spike/After Potential Dynamics:** After firing, the potential resets to a resting value and often includes a refractory period where the neuron is less responsive. #### Fractional Calculus in Neurons The extension into fractional calculus introduces memory and history effects: - **Fractional Derivative:** Adds the capability to model history-dependent processes, useful for capturing long-range temporal dependencies and memory effects in neuronal firing that traditional integer-order models may miss. This aspect helps model how past inputs or states influence current neuronal behavior much like in real biological neurons. - **Memory Effects:** The fractional term alters the integration process so that past membrane potential values (voltage) contribute over time, affecting the current integration state. This is biologically inspired by evidence of memory effects in neuronal processing, such as synaptic plasticity that involves long-term potentiation or depression of synaptic weights. #### Important Parameters in the Model - **Noise and Input Current (Iinj):** Imitates synaptic inputs or other external stimuli, reflecting biological inputs a neuron might receive. - **Time Constants and Resistances:** These parameters (e.g., `TauM`, `Rm`) control the dynamics of membrane potential decay and are crucial for proper temporal integration. - **Refractory Period:** Models the neuron's recovery period post-spike, during which it cannot fire again, accurately mimicking post-spike hyperpolarization. #### Biological Inhibitory Mechanism The model includes an inhibitory feedback component (`Inhib`), which could represent various biological processes such as synaptic inhibition or intrinsic mechanisms in neurons that affect spiking behavior by reducing the likelihood of firing after certain conditions are met. Overall, this approach attempts to align computationally modeled neuronal dynamics with the biologically observed phenomena, particularly those involving historical dependencies and feedback mechanisms, to produce a more robust and biologically plausible model of single-neuron activity.