\documentclass[11pt]{article}

\usepackage{amsmath,mathpazo,graphicx,url,doi}

\usepackage[a4paper,margin=2cm]{geometry}

\usepackage{xspace}
\newcommand{\betatwo}{$\beta 2$\xspace}

\usepackage{pdfpages}
\usepackage{setspace}

%% Switch to numeric style by passing [numbers]
%%\usepackage[numbers]{natbib}
\usepackage[]{natbib}

\begin{document}

\doublespacing

\newcommand{\um}{\ensuremath{\mu \text{m}}\xspace}
%%\newcommand{\ournote}[1]{\marginpar{\textbf{#1}}}

\title{Modeling developmental patterns of spontaneous activity}

\author{Julijana Gjorgjieva$^1$, Stephen J. Eglen$^{1,\dagger}$}
\date{$ $Revision: 1.39 $ $ \today}
\maketitle

\noindent $^{1}$ 
Cambridge Computational Biology Institute\\
Department of Applied Mathematics and Theoretical Physics\\
University of Cambridge\\
Wilberforce Road, Cambridge CB3 0WA, UK

\vspace*{2mm}


\noindent $\dagger$
Corresponding author.\\
\noindent Phone: +44 (0) 1223 765761

\noindent Email: S.J.Eglen@damtp.cam.ac.uk



\vspace*{1cm}
\noindent To appear in \textit{Current Opinion in
  Neurobiology (2011)} 

\noindent \renewcommand{\doitext}{DOI:}
\doi{10.1016/j.conb.2011.05.015}

\vspace*{2cm}
\subsection*{Abbreviations}
\begin{tabular}{ll}
(s)AHP & (slow) afterhyperpolarization\\
LGN & lateral geniculate nucleus\\
MEA & multi-electrode array\\
RGC & retinal ganglion cell\\
SAC & starburst amacrine cell\\
\end{tabular}


\clearpage

%% These allow us to annotate the bibliography
\newcommand{\refbb}{\\ $\bullet \bullet$ }
\newcommand{\refb}{\\  $\bullet$ }




%% Max 120 words for abstract.
\begin{abstract}
  Spontaneous activity is found in many regions of the developing
  nervous system; such activity is thought to be instructive for
  guiding developmental processes.  In particular, the developing
  retina generates correlated patterns of activity known as retinal
  waves.  We review the main theoretical models that have been
  developed to study the mechanisms for generation and propagation of
  retinal waves.  Much of the progress in this field has been due to
  the close interaction between experimentalists and theorists in
  analyzing and modeling spontaneous activity.  We conclude by
  describing spontaneous activity models in other systems and
  suggestions for future modeling work.
\end{abstract}

\section*{Introduction}

Neurons in the developing retina generate spontaneous activity in the
absence of sensory input.  This activity is highly structured, often
generating waves of activity that travel across the retinal ganglion
cell layer, followed by long periods of silence.  These retinal waves
generate distance-dependent correlations among retinal ganglion cells
(RGCs).  The initial \textit{in vivo} recordings of correlated
activity in embryonic rat \citep{maffei90} have subsequently led to a
large number of studies in which the pharmacological mechanisms
underlying wave generation and propagation have been studied
\citep{wong99}.  Retinal waves have been observed in a wide range of
species including mouse, rat, ferret, turtle and rabbit
\citep{sernagor2006}, with the one exception in \emph{Xenopus}
tadpoles \citep{demas2011}. We review the general mechanisms 
responsible for wave generation and propagation,
which have been the key assumptions for building theoretical models
of retinal waves. Furthermore, spontaneous neural activity
is present not only in developing retina, but in many other parts of
the nervous system, including hippocampus, spinal cord and cochlea
\citep{blankenship2010,odonovan1999}. Some of the known principles for
retinal waves may also be appropriate for modeling spontaneous 
activity in these other systems.

\section*{Theoretical models of retinal waves}

Over the years, several studies have proposed theoretical models to
account for the detailed spatio-temporal properties of retinal waves.
In two of the four models \citep{feller97, hennig2009}, 
there has been a very close relationship
between modeling and experiments, with models published
together with new experimental data.  We believe such close
interactions between theory and experiments are invaluable in
furthering our understanding of the neural circuits underlying
wave propagation.

The first model of retinal wave activity was published in 1994
\citep{burgi94,burgi94c}, at a time when relatively little was known
about wave generation.  As more experimental data has appeared,
the theoretical models have become more tightly constrained
\citep{feller97, godfrey2007, hennig2009}.  In particular,
two essential findings were the key role of cholinergic
neurotransmission \citep{feller1996,sernagor1996} and the refractory
mechanism \citep{zheng2006_ahp} in starburst amacrine cells.

There are three main mechanisms to consider when modeling spontaneous
activity (Figure~1): 

\begin{enumerate}
\item How do waves \textbf{start}?  Given that there is no external
  input (e.g. from visual stimulation), there must be some intrinsic
  mechanism by which neurons become active.


\item How do waves \textbf{propagate}?  Once some neurons become
  spontaneously active, how do they excite neighboring neurons?

\item How do waves \textbf{stop}?  Once a wave becomes initiated, does
  it continue traveling until it spreads over the entire retina, or is
  the activity spatially-restricted?

\end{enumerate}

Before addressing these mechanisms, we first summarize the modeling
frameworks used.  We refer to each model by the name of the primary
researcher that developed the model: Burgi \citep{burgi94}, Butts
\citep{feller97}, Godfrey \citep{godfrey2007} and Hennig
\citep{hennig2009}.  Table 1 compares the main features of these
models.  A longer review of some of these models, describing
underlying mechanisms in detail, has been undertaken elsewhere
\citep{godfrey2009_review}.


\subsection*{Modeling framework and connectivity}

Individual neurons have been modeled as either
relatively abstract activation units, integrate and fire neurons, or
detailed conductance models.  Before the primary role of cholinergic
neurotransmission upon retinal wave generation was discovered, the
Burgi model used a single layer of retinal ganglion cells (RGCs).
After starburst amacrine cells (SACs) were highlighted as the source
of cholinergic-mediated activity, subsequent models included
populations of SACs connected to RGCs \citep{feller97, butts1999}.
Within the Butts model, the RGC layer was needed to `read out' the
underlying activity in the SACs.  After that, the Godfrey model showed
that only an individual layer of neurons (SACs) was required, which
has since been adopted by later models.

RGCs have been modeled either as receiving synaptic input from SACs
(Butts) or without synaptic inputs but with diffusion of ions
mediating wave propagation (Burgi).  By contrast, in the models which
implement only a SAC layer, SACs have large
overlapping dendritic arbors that form an extensive network of
synaptic connections (Godfrey and Hennig).


\subsection*{Initiation}
Most models assume that waves initiate by the coincident activation of
sufficient numbers of SACs. One exception is the earliest model by
Burgi, which requires external input to RGCs.  In the other models
(Butts, Godfrey and Hennig) two processes contribute to the initiation
of a wave: (1) SACs spontaneously depolarize, and (2) SACs receive
excitatory input from other neighboring SACs. In the Butts model, each
SAC spontaneously depolarises with a fixed probability. In contrast,
in the Godfrey model SACs spontaneously depolarise deterministically,
either when the cell's threshold for activation decays to zero or when
the level of excitation of the cell exceeds the current threshold.
Hennig modeled wave initiation by injecting cell-driven noise into
each amacrine cells, whose source has not been determined.  The amount
of noise was determined to match recent experimental measurements of
the probability of spontaneous depolarization of SACs in rabbit
\citep{zheng2006_ahp}.

\subsection*{Propagation}
In the absence of synaptic connections, the Burgi model assumed
potassium released by one RGC could diffuse locally to excite
neighboring RGCs.  In the remaining models, activity propagated among
SACs via the lateral synaptic connectivity of SACs.  This connectivity
was fixed in the Butts model, decayed with distance between pairs of
SACs (Hennig model) or depended on the area of dendritic
overlap (Godfrey model).
 
In the Butts model, activity also propagated from SACs to RGCs via
synaptic connections.  RGCs fired only if sufficient neighboring SACs
were coincident, and thus provided a filter to prevent small numbers
of active SACs generating waves in the ganglion cell layer.

\subsection*{Refractoriness}
To prevent the continued activation of neurons, after firing for a certain
period neurons enter a refractory period.  Butts used a distribution
of refractory periods with a mean set to the peak of the interwave
interval distribution. The refractory periods of each SAC had to be significantly variable to prevent
synchronization of activity across the network. Godfrey and Hennig
followed subsequent experimental evidence that longer and more intense
SAC depolarizations increased the duration of the slow
afterhyperpolarizing potential (sAHP) of the SACs
\citep{zheng2006_ahp}.  Both models assumed that the refractory period
of an amacrine cell was inversely related to the duration of its
previous depolarisation.  However, the two models implemented
different assumptions to desynchronize the network.  Godfrey's model
implemented additional nonlinearities to boost differences in
refractory times between the center and the boundaries of a wave; thus
the decay from refractory state was nonlinear.  Hennig's model assumed
activation of the sAHP via a fast and a slow timescale, initiated by
autonomous bursts and bursts which occur as parts of waves,
respectively.  Thus, despite the linear decay from refractoriness, the
nonlinearity of the activation led to desynchronization of the
network.  

These models therefore propose two different mechanisms for wave
variability. (1) In the Godfrey and Hennig models, frequent
spontaneous SAC depolarizations generate wave variability, with
subsequent refractoriness (sAHP with a constant timescale) limiting
wave propagation.  Thus, SACs that are spontaneously active between
waves become less excitable, limiting their ability to depolarize in
the near future and participate in a wave.  (2) In the Butts model,
variability in the sAHP timescale limits wave propagation. If 
only \textit{relatively} infrequent spontaneous SAC
depolarizations were implemented in the model, SACs would rarely be refractory and waves would
propagate across the entire retina. By making the sAHP timescale
highly variable, SACs stop being refractory at different times,
thus generating variable wave trajectories.


\subsection*{Evaluation}

The current retinal wave models also differ in their ability to
capture experimentally-observed distributions of event sizes, for
instance, wave size and interburst intervals.  Indeed, the output from
several of these models has been carefully compared to different
experimental data.  Both the Butts and Godfrey models were based on
calcium imaging data.  The Godfrey model can account for retinal waves
from a range of species and developmental ages under different
parameters.  The Butts model was based on calcium imaging data in
ferret, and likewise identified parameter regimes for generating
different types of waves (``small'' vs ``large'' vs ``hyperactive'')
\citep{butts1999}.  The Hennig model perhaps provides the most
detailed quantitative comparison of simulated and observed waves from
MEA recordings in turtle and mouse retina.  In particular, the model
can recreate the power-law like nature of retinal wave properties
including wave duration and wave size.  (See below for
  alternative models that generate power-law distributions of activity
  size and duration.)  The Hennig model has not yet been compared to
calcium imaging recordings, but it is likely to also reproduce this
type of data.

\section*{Spontaneous activity in other systems}
%
Spontaneous activity occurs in many areas during development and
adulthood. Like retinal SACs, other neurons known as pacemakers form
networks which can initiate spontaneous activity \citep{strata97,
  lischalk09, conhaim10}. 
While the amacrine cells which can initiate and propagate spontaneous
activity in the retina serve as an example of an intrinsic pacemaker
to the network, the initiation zones driving activity also can be
extrinsic to the network which propagates this activity, as seen in
the midbrain \citep{rockhill09}.

Retinal wave propagation has been modeled as a chaotic process on a
two-dimensional lattice, where the spatial locations of initiation and
propagation of the waves are mostly random. Other tissue where
spontaneous activity occurs has much more stereotyped patterns of
initiation and propagation. In the developing cortex, for instance,
spontaneous waves are always initiated in two specific parts of an
\emph{in-vitro} slice and are localized to specific parts of the slice
\citep{lischalk09, conhaim10}.  The waves can vary in the spatial
extent of the propagation and the timing between events, but not in
the direction of propagation. These events have yet to
be modeled and analyzed for the information content they may carry
for the developing cortex.

Spontaneous activity is also observed in motor networks, of which the
developing spinal cord has received most modeling attention. The
earliest models used a similar mechanism as for the generation of
retinal waves, activity-dependent depression \citep{tabak00}, and
could capture the basic periodic patterns of activity.  Newer models
can explain more complex patterns regarding the duration of the active
and quiet periods of the network \citep{tabak10}. Even though this
system is quite simple, it provides another good example of linking
theory and experiment.  It also demonstrates that different
  systems use a range of mechanisms to generate activity, though
  synaptic depression is not thought to be involved in generating
  retinal waves \citep{zheng2006_ahp}.



Other studies have focused on building more abstract networks and
understanding the propagation of activity in these networks as a
function of connectivity, synaptic strength and intrinsic dynamics of
the network neurons.  Spontaneous activity
  propagation in adult cortical cultures can be described with
  equations that govern avalanches \citep{beggs03}. In these
avalanches, events emerge as the activation of one unit initiates the
successive activation of other units causing a cascade which
propagates through the entire system.  The spatial and temporal
distributions of these avalanches show no characteristic scale, and
have been fit to a power law with an exponent of
$-3/2$. Hennig et al.~\citep{hennig2009} found such scale-free
  distributions in retinal waves, such that waves occur in all sizes,
  including frequent waves mediated by a few SACs
  and rare events spreading over the entire retina.  However, unlike
  previous models, the Hennig model provided a biophysical mechanism
  for the generation of scale-free distributions through sAHP-mediated
  desynchronization.

The studies by Beggs and Plenz~\citep{beggs03} proposed that the 
power-law distributed event sizes could be explained 
with a theory for a critical branching process with a branching
parameter close to the critical value of 1 (at which activity does not
die out nor explode in the network, but it propagates in exactly a
one-to-one manner).  Such approaches are useful because they may
answer the question ``what is spontaneous activity good for?'' since
they suggest that these networks optimize information transmission.
For instance, in the simulated cortical networks which could propagate
avalanches, Beggs and Plenz \citep{beggs03} computed the information
between the input stimulus and the output response as the difference
of entropies (the entropy of the entire response set minus the entropy
of a response given a stimulus). Thus, for networks with a branching
parameter close to 1, the information was maximized, meaning that the
response of the network carried the most information about the
presented stimulus, compared to networks with branching parameters
smaller or larger than 1. A different approach is taken by studies
which examine the conditions for activity propagation or for the
emergence of synchrony in a network (reviewed in \citep{kumar10}), but
these are often criticized for having little experimental application.

We have aimed to understand the generation of retinal waves
because of their utility in synaptic refinements and retinotopic
organization in higher visual areas.  However, the theoretical work on
activity propagation, information transmission and neural coding has
developed new tools which can be used to understand not only the
mechanisms but also the computations performed by propagating activity
in networks.




\subsection*{Future directions}

Modeling of retinal waves has so far been quite successful, partly
because of the close interaction between theorists and
experimentalists.  However, there is still much modeling work to be
done to further our understanding of the development of neuronal
circuitry.  We close by mentioning some theoretical issues likely to
be of interest in the coming years:

\begin{enumerate}

\item With the exception of \citep{godfrey2007}, most modeling work
  has focused on waves driven by cholinergic circuitry.  Waves are
  thought to be generated by different mechanisms across development,
  from gap-junctions (stage 1), cholinergic neurotransmission (stage
  2), and glutamatergic neurotransmission (stage 3)
  \citep{huberman2008_arn,syed2004}.  How can mechanistic models
  account for stage 3 waves, given that there is currently no evidence
  for a mechanism to generate a refractory period?

\item Stage 2 wave properties can vary considerably, e.g. from P1 to
  P5 in mouse \citep{hennig2009}.  Recent results from rabbit retina
  \citep{famiglietti2010} suggest the effective cholinergic network
  grows rapidly over the first two postnatal weeks.  How do simulated
   wave patterns change as more SACs are added throughout
   the cholinergic network?

\item Most experimental observations of retinal waves have been
  limited to recording from a small fraction of the retinal area.  In
  such small recordings, boundary effects may dominate.
  Higher-density MEAs now allow us to see finer structure within
  retinal waves across larger areas.  For example, recent recordings
  from a 512-electrode array suggest that waves may preferentially
  travel roughly along one retinal axis
  \citep{stafford2009}. Likewise, a new 4096-electrode array
  \citep{berdondini2009} opens the possibility for recording across
  the entire neonatal mouse retina \cite{sernagor2011}.  When the
  models are scaled up to simulate larger portions of the retina,
  hence reducing border effects, will they replicate the observed
  patterns?

\item When the cholinergic network is genetically perturbed, waves
  disappear but resume a few days later \cite{stacy2005}.  Such
  homeostatic recovery of activity is widely seen in developmental
  contexts \citep{odonovan1999,blankenship2010}.  Abnormal activity
  patterns are also seen in mutant mice
  \citep{demas2006,stafford2009}.  What extra mechanisms are required
  within the models to generate these perturbed patterns?  

\end{enumerate}

These theoretical models can help evaluate
hypotheses for the generation of retinal waves.  Their simulated wave
patterns can also be used in other models
\citep{elliott1999,godfrey2009,thivierge2009_nn} to explore how
spontaneous neural activity refines neural connectivity
\citep{huberman2008_arn,feller2009_nd,chalupa2009_nd}.  (Note that
sometimes, rather than simulate spontaneous activity,
experimentally-observed spike trains could be input directly into
models to assess their instructional properties
\citep{lee2002,gjorgjieva2009}.)  However, spontaneous activity is
essential for many other developmental processes such as motor axon
guidance, expression of guidance molecules and neurotransmitter
specification \citep{blankenship2010}.  We hope that in the future
theoretical modeling can contribute to elucidating the key
features of spontaneous activity in these processes.


\subsection*{Acknowledgements}

We thank Matthias Hennig for comments on the manuscript.  This work is
supported by a Wellcome Trust programme grant (083205/B/07/Z).  JG was
partially supported by the McKnight Foundation.

% Start-Of-Trailer

\newpage
%%\bibliographystyle{jneurosc}i
\bibliographystyle{unsrt}
\bibliography{coin}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Figures at end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\pagestyle{empty}


 \begin{figure}
   \centering
   \includegraphics[width=15cm]{figure_coin}
   \caption{Schematic of wave generation.  Each circle represents an
     individual neuron.  At any time a neuron
     is quiet, firing or refractory.  Individual neurons may fire
     spontaneously independent of each other (neurons marked A);
     co-activation of e.g. two neighboring neurons, as at the bottom,
     may be sufficient to cause a wave to propagate to neighboring
     regions (A $\rightarrow$ B $\rightarrow$ C $\rightarrow$ D).  The
     wave stops when it can no longer recruit neurons that are
     still in their refractory period (marked with x).}
    \label{fig:cartoon}
 \end{figure}

\clearpage
%% Could put this into the main article itself; source currently in
%% waves_table.tex 
\includepdf{waves_table.pdf}
\end{document}







% LocalWords:  presynaptic postsynaptic STDP BTDP Eglen Julijana Gjorgjieva LGN
% LocalWords:  retinotopic tectum RGC RGCs ephrin geniculate Hebbian starburst
% LocalWords:  hippocampus neuromuscular retinogeniculate acetylcholine Xenopus
% LocalWords:  amacrines cholinergic retinotopy excitatory correlational axonal
% LocalWords:  glutamatergic potentiation olfaction tonotopic Semaphorin cAMP
% LocalWords:  repellant colliculus tectal rostral potentiated pre Eph antennal
% LocalWords:  glomeruli ORN glomerular amacrine Hennig neurotransmission
% LocalWords:  Burgi SAC SACs dendritic
