# Biological Basis of the Computational Model The provided code represents a computational model of a network of neurons, specifically inspired by leech-heart neurons. This model incorporates elements of fractional-order calculus, which introduces memory effects into the neuronal dynamics. The key components of the biological system being modeled and their relevance to the code are outlined below: ## Neuron Model - **Leech-Heart Neurons**: This neuronal model is based on leech-heart excitability dynamics. The leech, an invertebrate, has been extensively studied for understanding basic neuronal mechanisms. Leech-heart neurons are known for their rhythmic activity, which is a form of bursting oscillations fundamental to understanding basic electrophysiological processes. ## Key Biological Components - **Membrane Voltage (u)**: In the context of the model, `u` represents the membrane voltage of neurons. This variable is central to the code's representation of information processing in neurons, where changes in membrane voltage represent the potential for signal transmission within and between cells. - **Gating Variables (V and w)**: The model uses variables `V` and `w` that are analogous to gating variables in biological neuron models. These variables control the opening and closing of ion channels, fundamentally influencing the flow of ions across the neuron's membrane. - **Ion Dynamics**: The model describes ion channel dynamics using equations that involve exponential functions. The use of sigmoidal functions is typical in modeling ionic currents controlled by voltage-sensitive gates. ## Fractional Order Modeling - **Memory Effects**: The model includes fractional-order derivatives, which account for memory effects in neuronal dynamics. This is biologically relevant as biological tissues can exhibit properties that are dependent on past states, not just current states. The fractional-order aspect introduces non-locality in time, reflecting a more realistic description of neuronal processes. ## Network Dynamics - **Network of Neurons**: The model represents a network size of 100 neurons, with coupling between these neurons modeled through a connectivity matrix (`a`). Coupling represents synaptic interactions between neurons, which in a biological context account for neuronal communication and synchronization. - **Oscillatory and Excitable Neurons**: The model includes a subset of oscillatory neurons (`M=60`). This differentiation is crucial for simulating interactions between different types of neurons, as seen in biological systems where specific neuron types have distinct roles, such as excitability and rhythmic bursting. ## Modeling Objectives The primary objective of this computational model is to study the emergence of bursting behavior within a network of neurons. Bursting is a critical phenomenon in many neural systems, contributing to functions such as signal modulation and synchrony. Understanding the underlying mechanisms of bursting in a network provides insights into neurological oscillations and dysfunctions. In summary, this computational model captures the complexity of neuronal interactions in a network, with a focus on dynamics relevant to leech-heart neurons, through fractional calculus to simulate memory effects and bursting behavior.