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%\footnote{DELETE THIS FOOTNOTE BEFORE SUBMISSION.  In the \TeX \, source code there is a toggle/option to produce a Teacher Version or a Student Version by ``carving'' out with  $\backslash$teacher\{\} that material which should only appear in the Teacher version. See the source code.} 
\else STUDENT VERSION \fi\\Coincidence detection in the integrate-and-fire neuron}





\markboth{Coincidence detection in the integrate-and-fire neuron}{Coincidence detection in the integrate-and-fire neuron}

\author{Joshua Goldwyn\\
Swarthmore College\\
Swarthmore PA  19096 USA\\
jgoldwyn@swarthmore.edu
}

\makeStitlePDFLaTex


\begin{abstract}
In this activity students will study a linear, first order, one-dimensional ordinary differential equation (ODE) and learn how it can be used to understand basics of neural dynamics.  The modeling framework is known in the mathematical neuroscience literature as the ``integrate-and-fire'' neuron.  The form of the model is equivalent to the linear ODE used to describe the RC circuit.  This activity assumes familiarity with solving linear first order ODEs, for instance using the integrating factor method.  Students are asked to set up and solve a homogeneous version of the equation and a nonhomogeneous version (constant forcing term).    Students are asked to set up ODEs (select initial values), solve the ODEs, and perform some related algebraic calculations.  Throughout, students are asked to interpret their results in light of biological experiments and biological terms that are explained in the activity.  There is a focus throughout on solving the ODE in the presence of unspecified parameters and interpreting how parameter values may affect response characteristics of biological neurons. Outcomes of the project include: improved skill setting up and solving linear first order ODEs (including with unspecified parameters) and demonstrating how mathematical models can improve understanding of dynamic biological systems.    There is no requirement or expectation that students have experience with biology or neuroscience.
\end{abstract}




\section*{SCENARIO DESCRIPTION}

The human brain is composed of billions of cells called {\it neurons}.  The electrical activity of neurons is the basis for our behaviors, perceptions, thoughts, dreams, and so much more.   Mathematical models of this electrical activity can help us understand how the brain works, how to treat brain diseases, and how to unravel the mysteries of cognition and consciousness.  

A starting point for understanding the brain is to model the activity of a single neuron.  The quantity that describes the electrical activity of neurons is the voltage difference between the inside and outside of the cell. 
 This voltage, typically referred to as {\it membrane potential} is expressed in units of millivolts (mV).  Figure~\ref{fig:vTrace} shows an example of the membrane potential of a neuron.    Striking features of the membrane potential of a neuron are the rapid and large swings in voltage (labelled as {\it spikes in $V(t)$}) and smaller fluctuations near a ``resting'' value of the voltage (labelled as  {\it $V(t)$ near rest}).  Since membrane potential is a quantity that changes over time, ordinary differential equations are a natural mathematical language for describing the time-course of $V(t)$.  Modeling $V(t)$ in all its detail requires systems of nonlinear differential equations~\cite{Izhikevich2010}, but in this activity you will learn that much about a neuron can be learned by describing $V(t)$ with a linear, first order, one-dimensional ordinary differential equation (ODE).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=12 cm]{vTrace.pdf}
\caption{Voltage in a neuron exhibiting large and sudden spikes and small fluctuations near rest.  This voltage time-course was recorded from a slice of a rat's brain taken from a region that is involved in the perception of smell (the olfactory bulb)~\cite{Desmaisons1999}.  }
\label{fig:vTrace}
\end{center}
\end{figure}

The membrane potential changes when electrically-charged particles, called {\it ions}, flow into and out of the cell.  For example, a flow of positively-charged ions into the cell will increase the membrane potential.  The idea behind most mathematical models of neurons is to treat flow of current into and out of a cell as you would the flow of current around an electrical circuit.   
An important example of a mathematical model that uses this electrical circuit idea to describe a neuron's voltage is the {\it integrate-and-fire model}~\cite{Brunel2007,Burkitt2006}.  It is one of the first models used to describe neural voltage and is still used by many researchers.
This model was created by assuming that the membrane of a cell acts the same as an electrical circuit composed of a  capacitor in series with a resistor.  
The ODE associated with this model is:
\begin{align}
\label{eq:RC}
C \frac{dV}{dt} = -\frac{1}{R}(V(t)-E) + I(t).
\end{align}
The following parameters are constants:  $C$ is the membrane capacitance (units: picoFarad), $R$ is the membrane resistance (units: megaOhm), and $E$ is the resting potential of the neuron (units: milliVolts).  The term  $I(t)$ is an input current (units: nanoAmps).  This term can represent, for example, current injected into the neuron by a neuroscientist who has inserted an electrode into the cell, or the natural flow of current between two neurons that are coupled together in the brain through a structure called a ``synapse.''



This equation can be simplified by introducing a new parameter $\tau = RC$ called the {\it membrane time constant} (units: milliseconds) and by measuring voltage relative to the resting potential $v(t) = V(t) - E$.  The resulting first order linear differential equation for membrane potential is
\begin{align}\setcounter{firstbib}{\value{enumiv}}
\label{eq:dvdt}
\tau \frac{dv}{dt} = - v(t) + RI(t) 
\end{align}

In this project you will study solutions to this equation and pay particular attention to the importance of the parameter $\tau$.  You will learn how it can be measured in real neurons and how the value of this parameter can determine how neurons respond to different types of inputs.

\subsection{Estimating $\tau$ from the voltage dynamics of neurons}

The membrane time constant is an important parameter that determines the dynamics of the neuron.  A way to measure $\tau$ in real neurons is to start an experiment with the membrane potential beginning at a value that is slightly above its resting value.  The neuroscientist can then measure $v(t)$ as it decays to 0~mV.  If the time-course of $v(t)$ returns to 0~mV in the shape of an exponentially-decaying curve, then the decay rate of this curve can used to estimate the membrane time constant $\tau$.
\begin{enumerate}[label={(\alph*)}] 
\item Set up and solve the initial value problem that would model this experiment.  Choose a few different values of $\tau$ and sketch the solution.  Typical values of $\tau$ in real neurons range from around 1 millisecond to 100 milliseconds. A database of values of $\tau$ for many different types of neurons can be found at \url{https://neuroelectro.org/ephys_prop/4/}.
%\subitem {\it Should I include some hints?  The initial voltage $v(0)$ is some non-zero value, students can call it $v_0$.  There is no input current being supplied to the cell so students should should let $I(t)=0$ for all $t$.}
\item Figure~\ref{fig:timeConstant} is a graph that was published in a scientific journal article~\cite{Scott2005}.  It shows the voltages in a real neuron measured by neuroscientists (thin black line) and the exponential curves that the neuroscientists fit to these data in order to estimate $\tau$ (thick black lines).  The membrane time constants estimated from these curves were approximately $0.4$~ms and $1.2$~ms.  Say which curve (``normal'' or ``DTX-K'') goes with which time constant ($0.4$~ms or $1.2$~ms).
%\subitem {\it Possible hint: students use graphing calculator or desmos to graph their solution from (a) with different values of $\tau$.}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=12 cm]{timeConstants.png}
\caption{Voltage in a neuron as it returns to rest in a control condition (``normal'') and after application of dendrotoxin-K (``DTX-K'').  These voltages were recorded from a neuron in a slice of a gerbil's brain taken from a region that is involved in the perception of sound (the medial superior olive)~\cite{Scott2005}. The scale bar at right represents a time interval of length 0.5~milliseconds.  The thicker black curves show the exponential curves that have been fit to the recorded voltage trace. }
\label{fig:timeConstant}
\end{center}
\end{figure}

\item The label ``DTX-K'' refers to a neurotoxin that the scientists applied to the neuron to change its membrane resistance (parameter $R$ in (\ref{eq:RC})).  Did the use of DTX-K increase or decrease membrane resistance? Explain your reasoning briefly (one sentence should be enough).
\end{enumerate}


\subsection{Threshold for spikes and the interspike interval}

The linear ODE in (\ref{eq:dvdt}) cannot produce the large and rapid changes in $V(t)$ that are labelled as {\it spikes} in Figure~\ref{fig:vTrace}.  Instead, users of this model define some constant voltage value to be the {\it spike threshold}, often denoted by $\theta$.  Any time $v(t)$ reaches this threshold value, the model neuron is said to have produced a spike and the voltage is immediately reset to the resting voltage $v=0$~mV.  This process of $v(t)$ approaching threshold and then spiking (or ``firing'') when $v(t)$ hits the threshold is why this model model is known as the {\it integrate-and-fire} neuron.

\begin{enumerate}[label={(\alph*)}] 

\item Consider the case of a neuron receiving a constant, positive input ($I(t) = I$) and set the spike threshold to be $\theta = 1$~mV.   Set up and solve an initial value problem in this case.
\item Suppose a spike has just occurred at time $t=0$.  Use your answer to (a) to find the time to the next spike.  You can denote the time to the next spike as $T$.  This quantity is known to neuroscientists as the {\it interspike interval}.   Your final answer should be an equation for $T$, expressed in terms of the  parameters $\tau$, $R$, and $I$.
\item How does the interspike interval depend on $\tau$?  In other words, does $T$ increase or decrease with $\tau$?
\item Based on your answer to (c) and your observations in Question 1, would you describe a neuron with a small membrane time constant as ``fast'' or ``slow''? 

\end{enumerate}


\subsection{Neural coincidence detection}
The billions of neurons in the brain exhibit a wide variety of dynamics and have a wide variety of ``jobs.''  As a result, neurons can have very different electrical properties from one another.  
Membrane time constants, for instance, are smaller than 1~ms in some neurons and larger than 100~ms in other neurons.  
%An interactive table that compiles time constants for many different neurons can be found at \url{https://neuroelectro.org/ephys_prop/4/}.

For this problem, you will think about neurons that have the specialized capability to generate spikes only when inputs arrive close together in time, and to not generate spikes when inputs do not arrive close together in time.  Neurons with this property are called {\it coincidence detectors}.  You will  use your understanding of  (\ref{eq:dvdt}) to predict whether coincidence detector neurons should have small or large membrane time constants.

Suppose $I(t)$ consists of two pulses of current that are infinitesimally-narrow in time.  Mathematically such instantaneous events can be described by something called the Dirac delta function.  We say the first current pulse occurs at time $t_1$ and the second occurs some time later at time $t_2$.  Each of these inputs causes the voltage of the neuron to instantaneously increase (``jump up'') by an amount that we will call $\Delta v$. 
Figure~\ref{fig:coincidentInput} shows examples of the voltage response to these inputs (solid black curve with two peaks).  In this example, a first input arrives at $t_1 = 0$~ms and increases the voltage by $\Delta v = 2/3$~mV and a second input of the same strength arrives at $t_2 = 1$~ms.  Thus the two inputs are separated by $t_2-t_1 = 1$~ms.  These inputs do not increase the voltage enough to bring $v(t)$ across the threshold (shown as a dashed black line at the top of the figure). If the inputs were closer together in time, then the voltage responses to each input may sum up and cross threshold.  The instantaneous input pulses described in this problem can be thought of as a simple way to model how neurons are connected to one another in the brain through structures called {\it synapses}.
% here can be represented mathematically as {\it delta functions} and 

\begin{figure}[htbp]
\begin{center}
%\includegraphics[width=.8\textwidth]{cd.pdf}
\includegraphics[width=.8\textwidth]{RevisedCD.pdf}
\caption{Voltage trace of the integrate-and-fire model in response to instantaneous inputs that displace voltage by an amount  $\Delta v = 2/3$~mV.  In this figure the first input is at $t_1=0$~ms and the second is at $t_2=1$~ms, resulting in a time difference of $t_2-t_1 = 1$~ms. Other parameters used for this figure: membrane time constant is $\tau = 1$~ms and spike threshold is $\theta=1$~mV (horizontal dashed line).}
\label{fig:coincidentInput}
\end{center}
\end{figure}

\begin{enumerate}[label={(\alph*)}] 

\item Set up and solve the initial value problem that models the voltage of a neuron immediately after the second of the two instantaneous inputs.  In other words, set up and solve the initial value problem to calculate $v(t_2)$.  To do this, you will first set up and solve an initial value problem that described $v(t)$ in the time interval $t_1 \leq t < t_2$.  Then you will add $\Delta v$ to your solution to get the value of $v$ after it has been instantaneously increased by the second input.  

 Your final answer will be an equation for $v(t)$ in terms of the parameters $t_2-t_1$ (the time between inputs), $\tau$ (the membrane time constant), and $\Delta v$ (the voltage increase caused by each input).  To simplify your calculations you can assume the first input arrives at $t_1 = 0$~ms.

\item Using your answer to (a), find an equation for the largest value of $t_2-t_1$ for which the neuron's voltage crosses threshold.  This quantity is called {\it coincidence detection window} of the neuron.  If inputs arrive closer together in time than that value, the neuron will spike (its voltage will cross threshold).  If inputs arrive further apart in time, voltage will remain below threshold.  
\item How does the coincidence detection time window depend on $\tau$? 
\item A good coincidence detector neuron should spike only when inputs arrive close together in time and not spike when inputs arrive far apart in time. Based on your calculations, do you predict that coincidence detector neurons are  ``fast'' or ``slow'' neurons (small or large time constants)?
\item Coincidence detectors can be found at several locations in the brain involved in the processing of sounds.  Two such regions are the {\it cochlear nucleus (ventral) octopus cell} and the {\it medial superior olive ventral cell}.  Explore the database of time constant measurements at \url{https://neuroelectro.org/ephys_prop/4/} and determine if the time constants of neurons in these regions match what you predicted in (c). Explain (briefly) what you find.  
\subitem {\it Tip: } Click the blue button ``view data in table form'' to search the table more easily.
\end{enumerate}

%\section*{Submission Version}
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% 
%

%Below is a set of References which appears in the Student Version. Note, there is a second set of References one can use for the Teacher Version.

\begin{thebibliography}{1}

\bibitem{Izhikevich2010}
Izhikevich, Eugene~M. 2010.
{\it Dynamical Systems in Neuroscience}.
Cambridge MA: MIT Press Ltd.

\bibitem{Brunel2007}
 Brunel, Nicolas and Mark C.~W. van Rossum. 2007.
 Lapicque's 1907 paper: from frogs to integrate-and-fire.
{\it Biological Cybernetics\/}. 97(5-6): 337--339.

\bibitem{Burkitt2006}
Burkitt, A.~N. 
2006. A review of the integrate-and-fire neuron model: I. homogeneous
  synaptic input. {\it Biological Cybernetics\/}. 95(1):1--19.

\bibitem{Desmaisons1999}
 Desmaisons, David, Jean-Didier Vincent, and Pierre-Marie Lledo. 1999.
Control of action potential timing by intrinsic subthreshold
  oscillations in olfactory bulb output neurons. {\it The Journal of Neuroscience\/}.  19(24): 10727--10737.


\bibitem{Scott2005}
Scott, L.~L. 2005.
Posthearing developmental refinement of temporal processing in
  principal neurons of the medial superior olive.
{\it Journal of Neuroscience\/}. 25(35): 7887--7895.

\setcounter{firstbib}{\value{enumiv}}

\end{thebibliography}


\teacher{
\section*{NOTES FOR TEACHERS}

\subsection*{Background and supplemental resources}
This activity is an introductory example of how differential equations are used to study the voltage dynamics of neurons.  There is extensive literature on this approach to modeling neurons including textbook treatments~\cite{Izhikevich2010}.  This activity is restricted to a first order linear ODE.  With a threshold condition incorporated, this model is known as a {\it linear} integrate-and-fire model. See~\cite{Burkitt2006} for a general review of the integrate-and-fire model. Numerous extensions of the model could be considered in later stages of an undergraduate ODE course including nonlinear versions of the integrate-and-fire model, e.g., the quadratic integrate-and-fire model and exponential integrate-and-fire model~\cite{Gerstner2014}.  There are also well-known nonlinear dynamical systems that capture the dynamics of excitability and spiking in neurons.  These include the celebrated Hodgkin-Huxley equations as well as related two-dimensional nonlinear systems that can be studied with phase plane techniques, e.g., the Fitzhugh-Nagumo equations and the Morris-Lecar model~\cite{Izhikevich2010}.

There is a specific focus in this activity on ``neural coincidence detection.'' This refers to the capacity of some neurons to respond (generate spikes) selectively depending on the relative timing of their inputs.  Coincidence detection is known to be important in specialized regions of the auditory system~\cite{Golding2012} and coincidence detector neurons have also been studied in the cortex~\cite{Roy2001}.   There have been several mathematical studies of neural coincidence detection using ODE models and dynamical systems techniques~\cite{Rinzel2013}.

\subsection*{Logistics}
This activity can be offered as an in-class activity, as a take-home project, or as a combination of the two.  A recommended strategy would be for the instructor to use some class time to introduce relevant biological vocabulary and concepts, introduce the integrate-and-fire model, and begin some calculations in Exercise 1.  The instructor can then use the following class period for students to complete the exercises.  Some or all of the exercises could also be assigned as take-home projects.  This project can be carried out in small groups (2-4 students) or assigned to individual students.  A shorter version of the activity would be to do problems 1 and 2 only.  In this case, the students would not be exposed to the idea of {\it neural coincidence detection} that is explored in problem 3.

Software and numerical computations are not required.  Students are asked to consider solutions to linear first order ODEs, so they may appreciate access to a graphing calculator or online graphing tool (such as desmos.com) if they are uncomfortable graphing these solutions by hand.

\subsection*{Solutions}

\begin{enumerate}

{\bf \item Estimating $\tau$ from the voltage dynamics of neurons}

\begin{enumerate}[label={(\alph*)}] 
\item Students must solve  $$ \tau \frac{dv}{dt} = -v $$ with some initial value, call it $v(0)=v_0$.  The solution to this initial value problem is
$$ v(t) = v_0e^{-t/\tau}$$

\item[]{\it Tip:} The problem could be made easier if the instructor specifies the initial value as part of the problem statement (either as a particular value, or as a parameter $v_0$). 

\item[]{\it Tip:} Students can be encouraged to graph their solution for different values of $\tau$, either by hand or with a calculator/software tool.

\item The slower decaying curve (``DTX-K'') has the larger time constant ($\tau = 1.2$~ms).  The faster decaying curve (``normal'') has the smaller time constant ($\tau = 0.4$~ms)

\item DTX-K led to larger membrane time constant (slower decay) and thus caused an increase in membrane resistance because the membrane time constant and resistance are related through the equation given in the project statement: $$ \tau = RC $$

\item[]{\it Tip:} The specific mechanism here is a block of Potassium-selective ion channels that reduces the conductance (increases the resistance) of the cell membrane.  Students can be told that the parameter $C$ (membrane capacitance) can be regarded as a constant.


\end{enumerate}

{\bf \item Threshold for spikes and the interspike interval}

\begin{enumerate}[label={(\alph*)}] 

\item Students must solve  $$ \tau \frac{dv}{dt} = -v + RI $$ with initial value $v(0) =0$.   The initial value is $v(0)=0$ because the problem states that the voltage is reset to its resting value after a spike. Using the method of integrating factors, the solution to this initial value problem can be found to be
$$ v(t) = RI(1- e^{-t/\tau}).$$

\item If $T$ is the time of the next spike, then set $v(T) =1 $ (the threshold value).  Students must apply this information to obtain an algebraic equation for $T$:
$$ 1 = RI(1- e^{-T/\tau})$$
which can be solved for $T$. The final answer is
$$ T = -\tau\ln(1-\frac{1}{RI})$$

\item[]{\it Tips:} To continue the theme of performing calculations in the face of unspecified parameter values, the threshold value could be said to be $\theta$, rather than assign the arbitrary value of 1~mV. 

\item   The logarithm term is negative for $I>1/R$, so the correct answer is that $T$ increases (linearly) with $\tau$. 

\item[]{\it Tips:} Astute students will notice this solution is negative for $I<0$ and and undefined for $I \leq 1/R$.  Students could be asked to describe what is happening in these situations.  

\item This question may generate some discussion. The expected answer is that small $\tau$ causes faster decay and shorter times between spikes, so ``fast'' neurons have small membrane time constant.


\end{enumerate}

{\bf \item Neural coincidence detection}
\begin{enumerate}[label={(\alph*)}] 
\item Initial value problem to solve is 
$$ \tau \frac{dv}{dt} = -v  \quad \mbox{ with }  \quad v(t_1) = \Delta v $$
notice this is a homogeneous equation (because inputs cause instantaneous displacements of $v(t)$
and voltage at time $t_2$ will be the solution to this equation plus an additional $\Delta v$ (due to the effect of the second input.

Solution is:
$$v(t_2) = \Delta v (1 + e^{-(t_2-t_1)/\tau}) $$

\item Set $v(t_2) = \theta$ and solve the result from (a) for $t_2 -t_1$:
$$t_2- t_1 = -\tau \ln\left(\frac{ \theta}{\Delta v}-1 \right)$$
\end{enumerate}

\item[]{\it Tips:} Some discussion and/or extra questions could be given as scaffolding for this result.  In particular:  Implicit in the problem set up is that two input (close together in time) should suffice for $v(t)$ to cross threshold but one input should not.  Thus the relevant range of $\Delta v$ relative to voltage threshold is  $$\theta/2 < \Delta v < \theta. $$  As a consequence, the argument of the logarithm is between 0 and 1 and the logarithm is negative-valued.

\item[]{\it Remarks on input functions:} The instantaneous inputs used in this problem are presented here informally.  More formally, they could be described using the Dirac delta function.  Other functional forms that can be used when modeling synaptic inputs include exponential functions, double exponential functions, and functions of the form $( t / \sigma)e^{-t/\sigma}$ where $\sigma$ would be a time scale of synaptic activation and decay (known as alpha-function in the mathematical neuroscience literature).   For further work on non-homogeneous problems, one could also consider how the membrane time constant parameter $\tau$ impacts the voltage response to sinusoidal inputs (transfer function / Fourier analysis).
 
\item Based on the previous result, and the accompanying explanation (see {\it Tips}), the correct result is that the coincidence detection window $t_2-t_1$ increases (linearly) with $\tau$.

\item Good coincidence detectors should have short coincidence detection windows (only spike when inputs are close together in time). Conclude: want $\tau$ small, which means want coincidence detector neurons to be {\it fast}.

\item Students will find it easier to search the database at the link \url{https://neuroelectro.org/ephys_prop/4/data/} but they should be encouraged to look at the table too to observe that these two cell types have submillisecond membrane time constants, whereas most other cells in this data table have time constants on the order of 10~ms.  These coincidence detector neurons in the auditory pathway are among the ``fastest'' neurons in the brain, highly specialized to extract temporal features of sound.  Some further discussion could be encouraged about why neurons in the auditory pathway specifically may be specialized for temporal precision and coincidence detection.  Conversation could mention temporal features of sound such as pitch, rhythm, sound onsets and offsets, sound location, etc.

\end{enumerate}


\subsection*{Descriptions of resources and links}

\begin{itemize}
\item Students are asked to view and interpret a figure extracted from a scientific paper (Figure 2 in this scenario, obtained from Figure 5C in~\cite{Scott2005}. 
\item Students are asked to navigate a database of neurophysiological measurements available at the website \url{https://neuroelectro.org/ephys_prop/4/}.
\end{itemize}


\subsection*{Assessment}

\begin{itemize}
\item The following problems assess a student's ability to set up and solve linear first order initial value problems:\\
1a, 2a, 3a
\item The following problems assess a student's ability to algebraically manipulate ODE solutions to obtain related quantities of interest:\\
2b, 3b
\item The following problems assess a student's ability to interpret parameter values in a mathematical model in relation to biological concepts:\\
1c, 2cd, 3cd
\item The following problems assess a student's ability to explore and understand data sets and graphs in relation to a mathematical model:\\
1b, 3e
\end{itemize}

\noindent A suggested assessment strategy is to assign a certain number of points to each skill.  Points can be assigned to each skill based on correctness, clarity, and completeness of written responses to each problem.


% Second set of References one can use for the Teacher Version.

\begin{thebibliography}{99}

\setcounter{enumiv}{\value{firstbib}}

\bibitem{Gerstner2014}
Gerstner, Wulfram, Werner~M. Kistler, Richard Naud, and Liam Paninski.
2014. {\it Neuronal Dynamics\/}. Cambridge UK:  Cambridge University Press.

\bibitem{Golding2012}
Golding, Nace~L.  and Donata Oertel. 2012.
Synaptic integration in dendrites: exceptional need for speed. {\it The Journal of Physiology\/}. 590(22): 5563--5569.


\bibitem{Roy2001}
Roy, Stephane~A.  and Kevin~D. Alloway. 2001. Coincidence detection or temporal integration? What the neurons in
  somatosensory cortex are doing. {\it The Journal of Neuroscience\/}. 21(7): 2462--2473

\bibitem{Rinzel2013}
 Rinzel, John and Gemma Huguet. 2013.
Nonlinear dynamics of neuronal excitability, oscillations, and
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