The provided code simulates a network of neurons using a computational model based on the Izhikevich neuron model, which is widely used for simulating large-scale networks due to its balance between biological realism and computational efficiency. Below are the key biological aspects being modeled: ### Izhikevich Neuron Model - **Membrane Potential Dynamics**: The Izhikevich model characterizes the dynamics of the membrane potential (`v`) and a recovery variable (`u`). The model is described by two coupled differential equations that account for the neuron's spiking behavior. - **Parameters**: - **Membrane Capacitance (`C`)**: Represents the ability of the neuron's membrane to store charge. - **Resting Membrane Potential (`vr`)**: The baseline membrane potential at rest. - **Reset Voltage (`vreset`)**: Voltage to which the membrane potential resets after a spike. - **Threshold (`vt`)** and **Peak Voltage (`vpeak`)**: Define the spike initiation and peak voltage, critical for generating action potentials. - **Recovery Variable (`u`)**: A synaptic adaptation variable that captures the neuron's ability to adapt its firing threshold over time. ### Synaptic Dynamics - **Synaptic Current and Variables**: The code includes variables to model synaptic dynamics through rise (`tr`) and decay (`td`) times, reflecting how synaptic currents change over time in response to spikes. - **Synaptic Integration**: Tracks post-synaptic currents (`IPSC`) and implements both single and double exponential integration to simulate synaptic decay and transmission delays, a biologically plausible feature of synapses. ### Network Connectivity - **Sparsity (`p`)**: Models the sparse connectivity typical of biological neural networks where each neuron connects to only a fraction of the total possible synapses. - **Static and Adaptive Synaptic Weights**: The code includes a static weight matrix (`OMEGA`) and an adaptive component (`E*BPhi'`) to mimic plasticity mechanisms that adjust synaptic strengths based on activity. ### Learning and Adaptation - **Real-Time Learning with RLS**: Implements a Recursive Least Squares (RLS) approach to adjust synaptic weights, allowing the network to learn and approximate target signals. This reflects biological learning mechanisms like synaptic plasticity (e.g., long-term potentiation). ### Target Functions and Outputs - **Van der Pol Oscillator**: The code sets up an external oscillating target signal, mimicking rhythmic patterns seen in neural activity (e.g., circadian rhythms). - **Output Approximant (`z`)**: Represents the network's attempt to approximate the target signal, similar to how neural circuits process and predict sensory information. ### Stability and Dynamics - **Eigenvalues Analysis**: Assesses the stability and dynamic changes of the network before and after learning, reflecting how neural circuits can reorganize structurally and functionally during learning processes. In summary, this code models the activity and learning of a simplified neural network, capturing essential features of neuronal membrane potential dynamics, synaptic interactions, network connectivity, and adaptative learning processes observed in biological neural systems. This simulation provides insights into how neurons might integrate synaptic inputs and adjust synaptic strengths to learn new information.