**2. Physiological brain networks**

#### **From individual neurons to dynamic neuronal networks**

In the following, embarking from the Hodgkin-Huxley neuronal membrane model we endeavour to create biologically realistic and computationally efficient models of spiking neurons and further on to generate a local spiking neuronal network of a 1000 excitatory and inhibitory neurons. Our aim is to study the behaviour of this simple network under different structural and functional constraints, and critical network parameters, in order to understand the rich network dynamics that emerge, and gain insight into the physiology of cortical neuronal networks and pathophysiology of seizures [12].

#### **2.1 Modeling the neuron**

#### **The Hodgkin-Huxley biological neuron model**

The Hodgkin-Huxley type models [13] represent the biophysical properties of cell membranes and ionic conductances (current flows) that help determine at any time the neuronal resting and action membrane potentials (for mathematical details [14]). The lipid bilayer is represented as a capacitance (Cm). Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (Ip) (**Figure 1**) [12, 15].

If the integration of Excitatory Postsynaptic Potentials (EPSP) and Inhibitory Postsynaptic Potentials (IPSP) at the long somatodendritic processes of pyramidal neurons (thousands of synaptic contacts) [16] is sufficient to shift the resting membrane potential at the axon hillock closer to threshold (around 55 mV, inside negative), voltage-gated fast Na-channels open up allowing an influx of Na<sup>+</sup> and depolarization current sufficient to turn the inside of the membrane positive, resulting in the generation of an action potential (up to +40 mV, the inside positive). The local reversal of the membrane potential during the upstroke makes the Na<sup>+</sup> channels rapidly turn into an inactivated (non-conducting absolute refractory) state, while different voltage-gated channels open up allowing together with leaky K<sup>+</sup> channels for the early repolarisation and late after-hyperpolarisation phases of the membrane potential (prolonged relative refractory state). This is a very simplified *integrate-and-fire model* of a neuron and accounts for the action potential generated in a neuron (**Figures 2** and **3a** and **b**) [19].

The processing of post-synaptic potentials is much more than a simple algebraic summation, most likely a geometrical (vectorial) spatiotemporal integration with very


#### **Figure 1.**

*The electronic circuit equivalent of the Hodgkin-Huxley biological neuronal model [14] from https://commons. wikimedia.org/wiki/File:Hodgkin-Huxley.svg By Krishnavedala via Wikimedia Commons—Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=21725464 with added-on fundamental equations for current flowing through the lipid bilayer (Ic), current through a given ion channel (Ii) and total current through the membrane (I) for a cell with potassium (K<sup>+</sup> ) and sodium (Na<sup>+</sup> ) channels. Vm is the membrane potential, Vi is the reversal potential of the* i*-th ion channel,* VK *and* VNa *are the potassium and sodium reversal potentials, respectively,* gK *and* gNa *are the potassium and sodium voltage-gated (nonlinear) conductances per unit area, respectively and* gl *and* Vl *are the leak (linear) conductance per unit area and leak reversal potential, respectively.*

#### **Figure 2.**

*(a) Main sodium and potassium conductance/current giving rise to the action membrane potential, reproduced under CC BY 4.0 from: Figure 6 of Johnson M & Chartier S (2017). Spike neural models (part I): The Hodgkin-Huxley model. The Quantitative Methods for Psychology [17]. (b) A graph of the sodium and potassium conductances (GNa and GK), their sum (Gm), and the membrane voltage (Vm) during a propagating nerve impulse, which is basically a numerical solution of the Eq. 4.32 published by Hodgkin AL & Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology [18].*

#### **Figure 3.**

*(a) Approximate plot of a typical action potential shows its various phases as the action potential passes a point on a cell membrane. The membrane potential starts out at approximately 70 mV at time zero. A stimulus is applied at time = 1 ms, which raises the membrane potential above 55 mV (the threshold potential). After the stimulus is applied, the membrane potential rapidly rises to a peak potential of +40 mV at time = 2 ms. Just as quickly, the potential then drops and overshoots to 90 mV at time = 3 ms, and finally the resting potential of 70 mV is reestablished at time = 5 ms. By Original by en:User:Chris 73, updated by en:User:Diberri, converted to SVG by tiZom*

*—Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2241513 (b) ion movements during an action potential, showing when channels open or close during the action potential. Bottom image shows the corresponding sodium channel states at specific points of the action potential. The diagram was created by: If Only and was retrieved online from: scioly.org/wiki/index.php/File:Image12.jpg (publicly available).*
