The provided code implements a computational model of neuronal dynamics using the Hodgkin-Huxley (HH) type formalism, a cornerstone of biophysical modeling of excitable cells. This model aims to capture the essential features of neuron electrical activity within a neural network, focusing on the interaction between excitatory and inhibitory neuronal populations. ### Biological Basis: #### Neuron Types: 1. **Excitatory Neurons**: These neurons are modeled as conducting positive charge, primarily facilitated by sodium (Na\(^+\)) and potassium (K\(^+\)) ions through voltage-gated ion channels. In this code, excitatory neurons follow the Hodgkin-Huxley dynamics with additional external and recurrent synaptic inputs. 2. **Inhibitory Neurons**: Similar in their HH structure but with some differences in ion conductance to reflect GABAergic (gamma-aminobutyric acid) function which typically results in hyperpolarization (inhibition). #### Ionic Currents and Conductances: - **Sodium (Na\(^+\)) Currents**: Sodium current mechanisms contribute to the depolarization phase of action potentials. The gating variables \(m\) and \(h\) regulate these channels and are integrated into the model equations. - **Potassium (K\(^+\)) Currents**: Potassium currents are crucial for repolarization and maintaining resting membrane potential. \(n\) is the primary gating variable controlling potassium channel opening. The model varies the potassium conductance (\(g_{ki}\)), particularly in inhibitory neurons, to analyze its effect on network dynamics. - **Leak Currents (Cl\(^-\), Na\(^+\), K\(^+\))**: Represent constant background ion movement that helps maintain resting potential and respond to external stimuli. These currents typically involve chloride (Cl\(^-\)) and minor passive sodium and potassium components represented by \(g_{cl}\), \(g_{kl}\), and \(g_{nal}\). #### Synaptic Dynamics: - Employs synaptic conductance models for interactions between neurons: - **Excitatory Synapses**: Governed by \(g_e\) and have a reversal potential \(v_e=55 \text{ mV}\), characteristic of excitatory post-synaptic potentials due to neurotransmitters like glutamate. - **Inhibitory Synapses**: Modeled by \(g_i\) with a reversal potential \(v_i=-90 \text{ mV}\), typical of inhibitory post-synaptic potentials from GABA. #### External Inputs: - Modeled using a Poisson distribution to simulate external stimuli as stochastic synaptic input, reflecting naturalistic sensory or cortical inputs the neurons might receive. #### Network Connectivity: - Includes sparse connectivity with specific coupling strengths reflecting biological synaptic strengths, attempting to mimic a sparsely connected cortical microcircuit. #### Simulation Aspects: - Examines neural responses across a range of external inputs and varying inhibitory potassium channel conductance. This could represent learning, adaptation, or sensitivity to different environmental conditions. ### Biological Relevance: This model is designed to study the dynamic behavior of a cortical network composed of excitatory and inhibitory neurons. By varying the potassium conductance in inhibitory neurons (g\(_{ki}\)) and observing its effects on network activity, researchers can investigate how inhibitory regulation affects overall brain function, potentially shedding light on phenomena such as oscillatory patterns, synchronization, and information processing in the brain. This is particularly relevant in understanding regulatory mechanisms at play in various cognitive processes and neurological disorders where neuronal excitability and inhibition balance is crucial.