Analysis of a map-based neuronal model

TL;DR

A neuronal non-smooth map-based model of subthreshold oscillations in neurons is proposed and studied and the behaviour of the model in a noisy medium is studied.

Abstract

Subthreshold oscillations in neurons are those oscillations which do not attain the critical value of the membrane's voltage needed for triggering an action potential (a spike). Their contribution to the forming of action potentials in neurons is a current field of research in biology. The present work approaches this subject using tools from mathematical modeling, more exactly, a neuronal non-smooth map-based model is proposed and studied. The behavior of the model in a noisy medium is also studied.

Figures (6)

• Figure 1: The four planar sets $D_1-D_4$ forming the domain of definition of the function $f_a$ for $a>0.$
• Figure 2: Dynamics of the map \ref{['ec1']} for $a$ in a neighborhood of $a_{NS}.$ The parameters are $m=0.02,$$s=1.1$ and a) $a=2$ (left), b) $a=a_{NS}=2.0852$ and c) $a=2.1$ (right). The closed stable curve exists for $a>a_{NS}.$
• Figure 3: Transition from silence to spiking activity through subthreshold oscillations as $s$ decreases in the system (\ref{['ec0a']}) for $m$ small, $m=0.02,$ and $a=2.1$
• Figure 4: Transition of the dynamics of the map \ref{['ec0a']} from silence to tonic spiking through subthreshold oscillations for $m$ small. The parameters are $a=2.1,$$m=0.02$ and a) $s=1.115$, b) $s=1.1$ and c) $s=1.09\ .$
• Figure 5: Noise applied to subthreshold oscillations in the map \ref{['ec0a']} for $m$ small gives rise to spikes. The parameters are $a=2.1,$$m=0.02,$$s=1.1$ and a) $\sigma=0.0001$, b) $\sigma=0.0004$ and c) $\sigma=0.004.$

Theorems & Definitions (9)

• Remark 2.1
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Theorem 4.1
• Remark 4.2
• Theorem 4.3
• Remark 4.4
• Remark 5.1