## Biological Basis of the Fractional LIF Model Code The provided code models the electrical activity of a neuron using a **Fractional Leaky Integrate-and-Fire (LIF) model**, which is a variation of the classical integrate-and-fire model commonly used in computational neuroscience. ### Key Biological Concepts 1. **Neuron as an Electrical Circuit:** - In the model, a neuron is represented as an RC (resistor-capacitor) circuit, where the membrane resistance (`Rm`) and the membrane time constant (`taum`) are key parameters. This approach reflects how neurons integrate incoming synaptic inputs over time. 2. **Membrane Potential (`V`):** - The model simulates changes in membrane potential (`V`) in response to an injected current (`Iinj`). The resting membrane potential (`vrest`), threshold potential (`vth`), and peak potential (`vpeak`) are specified to capture the neuron's excitability and spiking behavior. 3. **Action Potential and Firing:** - When the membrane potential reaches the threshold (`vth`), the neuron "spikes" or generates an action potential, and the code subsequently resets the potential to `vrest`. This mimics biological neurons firing action potentials when sufficiently stimulated. 4. **Fractional Calculus:** - The code uses a fractional order derivative (`alpha`), which introduces non-integer order dynamics. This captures complex, history-dependent processes in biological neurons that are not adequately modeled by standard integer-order kinetics, such as spike frequency adaptation and memory effects in neuronal firing. 5. **Spike Timing Adaptation:** - Spike Timing Adaptation refers to the progressive change in firing rate of a neuron when subjected to a constant stimulus. This model accounts for such behavior by incorporating fractional derivatives, representing how real neuronal membranes can show slower dynamics compared to simple exponential decay, which is typical in the classical LIF model. 6. **Refractory Period:** - The code includes an absolute refractory period (`refrac`) to simulate the time during which a neuron is unable to fire again following an action potential. This period is vital for ensuring realistic firing patterns and neuronal responses. 7. **Injected Current:** - The `Iinj` parameter simulates the input current that a neuron might receive in a biological system. The presence of noise (`Namp`) reflects biological reality where neural signals often deal with background synaptic activity and stochastic fluctuations. 8. **Initial Conditions:** - The initial membrane potential (`v0`) is set, and different initial conditions are explored. This captures the potential variability in resting states or prior conditions of neurons before synaptic inputs. ### Summary The code captures the electrical behavior of neurons by modeling membrane potential dynamics with an emphasis on spike timing adaptation, leveraging fractional derivatives to offer a nuanced representation of neural firing behavior beyond simple integrate-and-fire dynamics. This model highlights the adaptability and temporal complexity of neuronal responses, which are rooted in the biological properties of neuron membranes and cellular processes.