# The Soma

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The Soma is the spherical central part of the neuron It is typically about 20 um in diameter and contains a salty potassium solution inside the cell. It is separated by the neural membrane as described earlier. Contained inside the Soma are a number of very important structures called organells but that can be ignored for this study.

**The Soma**

Many thousands of Inputs to the Soma arrive from the Dendrites. Its single output is the Axon

The key feature of the Soma is its ability to initiate an Action Potential in the area labeled the Axon Hillock (shown below). Membranes of dendrites and the Soma body do not generate action potentials because they have very few voltage-gated sodium channels. Only the Axon is capable of generating action potentials and the process starts in the Hillock. This area is also called the spike-initiation zone.

## Action Potential

The Action Potential profile begins at the membrane rest potential (about - 60 mV) and rises rapidly in the depolarization phase toward 0 mV. It overshoots the mark and then reverses back down to the resting potential. It then overshoots that and moves to the hyperpolerized region and then recovers to the resting potential.

The model used here is of a neuron described in the book, *Dynamical Systems in Neuroscience*:* The Geometry of Excitability and Bursting* by Izhikevich, is not based upon biophysical parameters but is a simple model that faithfully reproduces all the neurocomputational dynamical features of the neuron. The model is a two-dimensional systems having a fast voltage variable *v *and a slower “recovery” variable *u,* which may describe activation of the K+ current or inactivation of the Na+ current or their combination. The simple model to reproduce spiking and bursting behavior of many known types of neurons is described by a pair of differential equations

$$\begin{array}{l}C\frac{dv}{dt}=k(v-{v}_{r})(v-{v}_{t})-u+S\\ and\\ \frac{du}{dt}=a\{b(v-{v}_{r})-u\}\\ if\\ V\ge {V}_{peak}\Rightarrow v\leftarrow c,u\leftarrow u+d\end{array}$$

Those two equations make up a model called a Quadratic Integrate-and-fire neuron. In the Neuron Simulater this is implemented (in the Delph language) with this code:

Vout := Vin + tau * (k * (Vin - Vr) * (Vin - Vt) - Uin + i) / CAP; Uout := Uin + tau * a * (b * (Vout - VRest) - Uin); if Vout >= vpeak then begin Vout := c; Vin := vpeak; // Reset the membrane Uout := Uout + d; // Reset the recovery variable end;

** Where:**

- Vout = The current calculated membrane potential
- Vin = The last membrane potential
- Uout = The current calculated recovery value
- Uin = The current calculated recovery value
- tau = Step time between samples
- Cap = 100 Membrane capacitance [pF]
- Vr = -60 Resting membrane potential [mV]
- Vt = -40 instantaneous threshold potential [mV]
- k = 0.7 Parameters used for RS type
- a = 0.03 ; Recovery time constant [ms]
- b = -2.0 Constant [pA/ohm]
- c = -50 Membrane voltage reset
- d = 100 ; For a neocortical pyramidal neuron

The numeric values for k,a,b,c, and d vary by the type of neuron being simulated.

### This Soma model contains the Cortex, Thalamus, Hippocampal, Basal Ganglia, Brain Stem, Entorhinal Cortex and Olfactory Bulb simulations. These all come directly from 'Dynamical Systems in Neuroscience. by Eugene Izhikevich

Eugene Izhikevich founder and CEO Brain Corp. San Diego