The code provided is based on the **Izhikevich neuron model**, a mathematical model of spiking neurons that is both computationally efficient and biologically plausible. This model is particularly useful in simulating large networks of neurons due to its simplicity and effectiveness in reproducing various neuronal firing patterns observed in real biological neurons. Here’s a breakdown of the biological basis and significance of the code: ### Excitatory and Inhibitory Neurons The model distinguishes between two types of neurons: **excitatory** and **inhibitory**, which form the basic building blocks of neuronal circuits in the brain. - **Excitatory Neurons (Ne=800):** These neurons release neurotransmitters that increase the likelihood of the receiving neuron generating an action potential. In the code, random variables (`re`) are used to introduce variability in the properties of individual excitatory neurons. - **Inhibitory Neurons (Ni=200):** These neurons release neurotransmitters that decrease the probability of the receiving neuron firing an action potential. The code utilizes a separate set of random variables (`ri`) to simulate the inherent diversity among inhibitory neurons. ### Izhikevich Model Parameters The model employs several parameters to mimic the dynamics of real neurons: - **'a' and 'b':** These parameters define the time scale of recovery (`a`) and the sensitivity of recovery (`b`) with respect to the subthreshold fluctuations (`v` or membrane potential). - For excitatory neurons, `a` is constant at 0.02, and `b` is 0.2. - Inhibitory neurons feature variable `a` and `b` values adjusted by the randomness (`ri`) to capture diversity. - **'c' and 'd':** These parameters determine the reset value of the membrane potential (`c`) and the after-spike reset of the recovery variable (`d`), reflecting biological reset and adaptation mechanisms. - The reset conditions are notably different for excitatory and inhibitory neurons, reflecting their distinct biological roles. ### Membrane Potential Dynamics - **Membrane Potential (`v`):** Set initially to -65 mV, this reflects the resting potential common in many neurons. Neurons fire an action potential when `v` exceeds the threshold of 30 mV. - **Recovery Variable (`u`):** Represents a recovery variable that provides feedback to the membrane potential, mimicking the dynamics of ion channel gating (e.g., potassium channels). ### Synaptic Connections and Inputs - **Synaptic Matrix (`S`):** Represents the network's synaptic connections, with positive weights for excitatory synapses and negative ones for inhibitory synapses. The structure is typically sparse, reflecting the sparse connectivity observed in real neural circuits. - **Input Current (`I`):** Simulates external thalamic input to the neurons, introducing noise (`randn`) to mirror the irregular nature of synaptic inputs from other brain regions. ### Spiking and Adaptation - **Spiking Mechanism:** Neurons spike when `v` reaches the threshold, and post-spike effects are captured by resetting `v` and adjusting `u`. - **Adaptation Dynamics:** The update of `u` simulates the slow processes of neuron adaptation, capturing how neurons adjust their activity over time in response to sustained inputs. ### Biological Relevance By employing simple yet biologically grounded parameters, the Izhikevich model enables the simulation of complex network dynamics similar to those observed in the brain, including oscillations, synchronization, and different neuronal firing patterns. This makes it an invaluable tool in computational neuroscience for understanding brain function and dysfunction at a systems level.