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Averaging Transformations of Synaptic Potentials on Networks


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The problem of the transformation of microscopic information to the macroscopic level is an intriguing challenge in computational neuroscience, but also of general mathematical importance. Here, a phenomenological mathematical model is introduced that simulates the internal information processing of brain compartments. Synaptic potentials are integrated over small number of realistically coupled neurons to obtain macroscopic quantities. The striatal complex, an important part of the basal ganglia circuit in the brain for regulating motor activity, has been investigated as an example for the validation of the model.
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Averaging Transformations of Synaptic Potentials on
H. R. Noori,a
aInterdisciplinary Center for Scientific Computing, University of Heidelberg, Im
Neuenheimer Feld 294, 69120 Heidelberg, Germany
The problem of the transformation of microscopic information to the macro-
scopic level is an intriguing challenge in computational neuroscience, but also
of general mathematical importance. Here, a phenomenological mathematical
model is introduced that simulates the internal information processing of brain
compartments. Synaptic potentials are integrated over small number of realis-
tically coupled neurons to obtain macroscopic quantities. The striatal complex,
an important part of the basal ganglia circuit in the brain for regulating motor
activity, has been investigated as an example for the validation of the model.
1. Introduction
The brain nuclei, as parts of complex brain networks, are comprised by
different types of inter- and projection neurons within subnetworks of highly
complex structure. Beside the anatomical formation, functional properties in
integration and processing of neural information become adapted during the
development of these regions until adolescence. In this study, adult brain nuclei
are considered after their final development, as part of information processing
Experiments on the topology of brain regions suggest that these structures
can be categorized as heterogeneous media. There are various mathematical
methods describing dynamical processes in heterogeneous media, such as asymp-
totic analysis, homogenization, and integral equations (Pavliotis and Stuart [17],
Haken [13], E and Engquist [8], Levin and Chao[15]). There are traditional net-
work approaches on the effects of properties such as the local connectivity of
neurons on the striatal function (Wickens et al. [21]). These studies concen-
trate merely on the network structure of a particular brain region e.g. striatum
and even then do not contain some important features such as the role of large
spiny neurons in the striatal function. Large spiny neurons in the striatum are
Corresponding author
Email address:, (H.
R. Noori)
Submitted Preprint June 18, 2009
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
cholinergic and are of great importance for processes involving addiction and
food intake (Rada et al. [18], [19], Avena et al. [1]).
In the present study, simplified but essential physiological processes have
been chosen as the biological foundation for the mathematical model. The
reader is advised to consider the appendix for a brief introduction into the
biological terminology.
The course of information processing in brain regions depends on the propa-
gation of synaptic potentials - which are averaged postsynaptic potentials - along
the subnetworks of these regions. Thus, understanding the physiology of this
integration process requires the comprehension of the neuronal architecture of
the brain region of interest. The regional subnetworks consist of different types
of neurons and characteristic connections among them that can be obtained
by experiments. In this study the intraneuronal connections are subdivided
into the global, and ultrastructural morphology. Golgi and immunohistochem-
ical methods provide optimal frameworks for obtaining the values of the two
morphological dimensions experimentally. The immunohistochemical studies
reveal the appearance of certain neurotransmitter systems in a structure; Golgi
studies provide information about the form and type of neurons expressing the
obtained neurotransmitter systems and on the morphological dimensions of the
brain region in terms of the characteristic connections between different types
of neurons in a brain region. In general, the structural topology of the brain
region is then obtained. The knowledge about the appearance of different types
of neurons and the intraneuronal synaptic interactions inside a nucleus allow us
to abstract its structure through networks.
One way of representing such networks is through their realization in con-
tinuous spaces. The attendance of a high amount of neurons supports the idea
of the appropriateness of such an approach for this purpose.
Because of the importance of the global and ultrastructural morphology for
the information integration process in a nucleus and the dependency of their
values on the spatial distribution of neurons, they shall be modelled as spatial
variables parametrized by the neuron (neurotransmitter) type.
To provide a general framework for the regional neuronal activity, we intro-
duce an integral operator averaging the information on two refinement scales.
The idea is to cover the space including the nucleus with copies of discrete
fundamental domains consisting of neuronal assemblies which we call n-cells.
Roughly spoken, n-cell patches are finite networks of neurons. The ultrastruc-
tural morphology of a nucleus in terms of the synaptic connectivities between
different neurons provides information on the local connectivities in the n-cells,
which are embedded as edges of neuronal network inside the n-cells. Within the
information on the adherence functions of n-cells (global morphology variables)
and the distribution of neurons, the averaging of synaptic potentials across the
network of nuclei is completed. The main advantage of this method is in its
appropriateness for applications on various brain regions.
Assembling a suitable number of n-cells by considering the large-scale inter-
action between the n-cells (adherence of n-cells), we obtain information on the
combined influence of several synaptic inputs on macroscopic neuronal activity
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
Figure 1: Composition of n-cells
(Fig. 1). One trivial example is the subthalamic nucleus which consists of only
one class of interneurons. In this case the number of n-cells under consideration
is one (Parent and Parent [16]).
The aim of this study is to present a mathematical technique for averaging
microscopic information along complex network structures such as the networks
of single brain regions. The technique is applied on the corpus striatum - a part
of the basal ganglia circuit which plays an important role in the regulation of
motor behaviour- and shown to mimic faithfully the qualitative oscillatory be-
haviour of this brain region. The general formulation of the averaging operator
suggest its suitability for other discrete multiscale problems, especially in the
research area of material sciences.
2. Model Statement
Statement:. The integration of the electric activity in a brain nucleus (infor-
mation processing) depends on the distribution of its comprising neurons, the
ultrastructural morphology as a local variable quantity and the global quantity
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
in terms of the composition of neuronal fundamental domains (n-cells), that
describes the topology of the brain region.
The concept of the n-cells is very essential for the understanding of the
averaging transformation. The idea is to path up a region with multiple copies
of discrete fundamental domains called n-cells. The n-cells compose the whole
brain region of interest by applying the global morphological properties and
contain the local morphological properties as finite graphs. Hence, they unify
the morphological dimensions into one notion.
Definition 1: A n-cell is a neuronal assembly represented by a finite graph
characterized by the local ultrastructural morphology. Within proper composi-
tion of the n-cells, as discrete fundamental domains, they patch up the region
of interest.
Let Ω R3a compact, path-connected set, be the brain region of interest.
Let ΩiΩ, iIbe a skeleton representation of the finite graph of a n-cell,
and Ibe the finite index set with #I= #{n-cells}. Furthermore, let xibe
the variable describing the network position of a neuron relative to the n-cell
i; and ythe continuity variable describing the spatial position of a neuron
embedded as points in Ω as a subset of the complete space R3.{z}denotes the
finite set of neurotransmitter families that appear in the brain region of interest.
By allocating any neuron to a neurotransmitter, zparametrizes the proposed
Definition 2: A synaptic potential uz
x(t) is the sum of all Hodgkin-Huxley
postsynaptic potentials of a neuron of type zat position xinside a n-cell.
iI , z , xi:uz
x: [0, T ]R.
To average the synaptic potentials uz
x(t) in n-cells, the local connectivity
function ψz: ΩiR+has to be characterized. By introduction of the global
connectivity function χi: Ω R+, describing the composition of n-cells, we will
cover Ω will n-cell patches. We write χ:= (χi) for the |I|-vector of composition
mappings. The normalized distribution function ρz: Ω [0,1] as the kernel of
the averaging operator will complete the information required for the integration
of synaptic potentials across the network of brain regions. These functions
(ψz, χi, ρz) are obtained from Golgi- and immunohistochemical studies.
The continuous realization of the network structure of the brain nuclei and
the idea of transforming the microscopic, synaptic information to the macro-
scopic level compatible with the internal structure of a brain region suggest the
construction of an averaging integral operator along networks.
Definition 3: A compartment is a multiple C:= (Ω, ρz, χ, ψz, I) which
characterizes a brain nucleus by its ultrastructural and global morphology.
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Definition 4: Let Cbe a compartment, then the averaged potential v(t) of
synaptic potentials uz
x(t) across the network of a single brain nucleus is repre-
sented by:
v(t) = X
|I|g(χ(y)) dy , [
g:R|I|R+is the proper averaging parameter estimated by immunohisto-
chemical experiments. It provides a combinatorial dimension for the composi-
tion of the n-cells in a compartment.
The universality of the concept of a compartment provides the possibility
of applying the averaging transformation of this type for a wide category of
network multiscale problems.
3. Numerical Simulations
The aim of this section is the numerical investigation of the dynamical be-
haviour of a compartment by known synaptic dynamics (in terms of electric
activity uz
x(t)) of single neurons. The corpus striatum is used as an non-trivial
example to illustrate the efficiency of the averaging operator method. The com-
plexity of the striatal structure on the one hand, and the importance of this brain
region as a substrate of the basal ganglia in the regulation of motor activity and
neurological diseases on the other hand, make it a proper representative for such
investigations. First, the main morphological and ultrastructural properties of
this compartment are summarized which are required to comprise the integral
function. These properties include the spatial distribution of the different neu-
ral populations (ρz), the neurochemical classification (z), and the intraneuronal
connections ((ψz, χi, g)). Then, the simulation results are discussed and com-
pared with the experiments.
A category of proper experiments for the validation of the numerical results
is represented by the local field potential (LFP) studies. Thereby, a signal is
recorded using a low impedance extracellular microelectrode, placed sufficiently
far from individual local neurons to prevent any particular cell from dominating
the electrophysiological signal. This signal is then low-pass filtered, cut off at
300Hz, to obtain the local field potential (LFP). The low impedance and
positioning of the electrode allows the activity of a large number of neurons to
contribute to the signal. The unfiltered signal reflects the sum of action poten-
tials from cells within approximately 50 350µm from the tip of the electrode
(Legatt et al. [14]).
The frequency of the LFP signals in corpus striatum will then be compared
with those appeared as the results of the numerical simulation of the averaging
transformation by given morphological properties.
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3.1. The Corpus Striatum
The neuronal populations of the striatum could be divided into four classes:
Spiny projection neurons (about 96% of the whole neural population) are GABAer-
gic neurons, which get external inputs from cortical areas, thalamus and sub-
stantia nigra pars compacta (Smith et al. [20]). These neurons also get inputs
from the dopaminergic, GABAergic and acetylcholinergic interneurons. Large
aspiny neurons (cholinergic), GABAergic interneurons and dopaminergic neu-
rons comprise the rest population of striatal neurons. The cholinergic interneu-
rons get external inputs from thalamus and substantia nigra pars compacta;
and internal inputs from the GABAergic interneurons. They project to the
dopaminergic and projection neurons. The action of the cholinergic interneu-
rons is assumed to have opposite effects on the action of the dopaminergic
neurons on the whole network. The cholinergic neurons inhibit the activity of
the projection neurons of the direct pathway (mostly D1system) and disinhibit
the activity of the indirect pathway neurons (D2system) (DiFiglia [7], Graybiel
et al. [12], DiFiglia et al. [5], DiFiglia and Carey [6], Gerfen [10], Betarbet et al.
[3], Flores-Hernandez et al. [9], Graveland et al. [11], Bennett and Wilson [2]).
A schematic representation of the microcircuitry of corpus striatum is shown in
Fig. 2.
MATLAB has been used to simulate the averaging across the network of
striatum. An overall number of 6400 neurons has been used for the simulation
of averaged potentials in striatum. The density distribution function ρis repre-
sented in Figure 3. The local and global connectivity function ψand χdefined
the necessary computational parameters based on Figure 2. We have induced
a single excitatory signal from cortical projection neurons to the cholinergic
neurons in striatum to simulate the simplest integration process.
It reveals that the activity transmission induced by a constantly activated
neurons is approximately radial and oscillatory. The activity of a single neuron,
forced by a constant input, produces activation-waves on the striatal populations
(Fig. 3).
Such neurons are simplified versions of the tonically active cholinergic in-
terneurons. Activations of neurons of other classes suggest similar activation
waves across the striatal populations. We observe that the simulated oscil-
lations of v(t) (50Hz) are qualitatively correlated with the LFP-studies of
basal ganglia which also suggest oscillations in control patients (Boraud et al.
[4]). The efficiency of the averaging transformation has been also investigated
for other brain compartments. In general, the averaged potentials of the com-
partments reveal oscillatory dynamical behaviour. For example, synchronized
gamma oscillations were observed in the globus pallidus. The simulation results
for the globus pallidus could be well reproduced by the reader using the aver-
aging transformation. Therefore, the representation of these results is omitted
in the present study.
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
Figure 2: Schematic ultrastructural morphology of the primate’s striatum, including the neu-
rochemical synaptology. The green/orange arrows denote inhibitory/excitatory afferents. The
dashed arrows are external efferents to the striatal neurons. Hereby, St1Aand St1Bdenote
the spiny projection neurons which project to Globus Pallidus externa and Globus Pallidus
interna. St2denotes the dopaminergic interneurons, St3the GABAergic interneurons, and
St4the acetylcholinergic neurons.
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
Figure 3: Left: The distribution of striatal interneurons based on experimental data. Right:
The oscillatory dynamics of the activity of striatal neurons by a single constant activation
with a realistic spatial distribution of the neurons (tonically active cholinergic neurons). The
light nodes denote active neurons.
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
4. Discussion
In conformance with LFP experiments, the multiscale averaging operator (1)
transforms the synaptic potentials along the network structure of brain region
such as corpus striatum to local field potentials of the same frequency range.
This represents a first step from the neuroscience at Hodgkin-Huxely dimension
to the system biological level of consideration. Although this work was initially
inspired by one of the most intriguing problems of computational neuroscience,
it appears to be appropriate for several other intriguing applications. The char-
acterization of the discrete fundamental domains in a complex network in terms
of n-cells which is one of the dominating concepts of this mathematical ap-
proach, provides the possibility of the multiscale transformations of information
along discrete structure that are continuously realized. Beside the advantages of
this model such as its agreement with electrophysiological experiments, further
computational investigations are needed to verify its validity for more complex
Acknowledgment.. The financial support by the Interdisciplinary Center for Sci-
entific Computing and the International Graduiertenkolleg 710 (DFG) and the
funding by National Genome Research Network are acknowledged.
Neurotransmitter -Neurotransmitters are the most common class of chem-
ical messengers in the nervous system;
Synapse-Chemical synapses are specialized junctions through which neu-
rons signal to each other and to non-neuronal cells such as those in muscles.
Chemical synapses allow neurons to form circuits within the central ner-
vous system. They are crucial to the biological computations that underlie
perception and thought;
Morphology-The term morphology in biology refers to form, structure and
configuration of an organism. This includes aspects of the outward appear-
ance (shape, structure, colour, pattern) as well as the form and structure
of the internal parts like bones and organs;
Ultrastructural morphology-Cell and tissue morphology in the electron-
microscope level;
Interneuron-An interneuron (also called relay neuron, association neuron
or local circuit neuron) is a multipolar neuron which connects afferent
neurons and efferent neurons in neural pathways. Like motor neurons,
interneuron cell bodies are always located in the central nervous system;
Golgi method-Golgi’s method is a nervous tissue staining technique;
Nature Precedings : hdl:10101/npre.2009.3348.1 : Posted 19 Jun 2009
Immunohistochemical studies -Immunohistochemistry or IHC refers to the
process of localizing proteins in cells of a tissue section exploiting the
principle of antibodies binding specifically to antigens in biological tissues;
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The effects of the acute ethanol consumption on the brain's neurochemistry are largely studied at the synaptic level. Here, the acute action of low dosages of ethanol, in terms of the inhibition of the glutamatergic system through antagonizing the N-methyl-D-asparate receptors, on the neurochemical oscillations along the neurocircuitry of the basal ganglia is investigated by mathematical models. Substantial alterations in the dynamical behaviour of the neurochemical oscillations after single administration of low dosages of ethanol have been observed. Significant dynamical changes in the gamma-aminobutyric acid and glutamate systems along the subthalamic-pallidal feedback loop and the dopamine system of the striatal complex suggest new perspectives in the understanding of the ethanol-induced motor dysfunctions.
This work represents an attempt to elucidate the neurochemical processes in the basal ganglia by mathematical modelling. The correlation between neurochemistry and electrophysiology has been used to construct a dynamical system based on the basal ganglia's network structure. Mathematical models were constructed for different physical scales to reformulate the neurochemical and electrophysiological behaviour from synapses up to multi-compartment systems. Transformation functions have been developed to transit between the different scales. We show through numerical simulations that this network produces oscillations in the electrical potentials as well as in neurotransmitter concentrations. In agreement with pharmacological experiments, a parameter sensitivity analysis reveals temporary changes in the neurochemical and electrophysiological systems after single exposure to antipsychotic drugs. This behaviour states the structural stability of the system. The correlation between the neurochemical dynamics and drug-induced behaviour provides the perspective for novel neurobiological hypotheses.
This introduction to multiscale methods gives readers a broad overview of the many uses and applications of the methods. The book begins by setting the theoretical foundations of the subject area, and moves on to develop a unified approach to the simplification of a wide range of problems which possess multiple scales, via perturbation expansions; differential equations and stochastic processes are studied in one unified framework. The book concludes with an overview of a range of theoretical tools used to justify the simplified models derived via the perturbation expansions. The presentation of the material is particularly suited to the range of mathematicians, scientists and engineers who want to exploit multiscale methods in applications. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable readers to build their own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter. All of the twenty-one chapters are supplemented with exercises.
Our understanding of the organization of the basal ganglia has advanced markedly over the last 10 years, mainly due to increased knowledge of their anatomical, neurochemical and physiological organization. These developments have led to a unifying model of the functional organization of the basal ganglia in both health and disease. The hypothesis is based on the so-called "direct" and "indirect" pathways of the flow of cortical information through the basal ganglia and has profoundly influenced the field of basal ganglia research, providing a framework ibr anatomical, physiological and clinical studies. The recent introduction of powerful techniques for the analysis of neuronal networks has led to further developments in our understanding of the basal ganglia. The objective of this commentary is to build upon the established model of the basal ganglia connectivity and review new anatomical findings that lead to the refinement of some aspects of the model. Four issues will be discussed. (1) The existence of several routes for the flow of cortical information along "indirect" pathways. (2) The synaptic convergence of information flowing through the "direct" and "indirect" pathways at the single-cell level in the basal ganglia output structures. (3) The convergence of functionally diverse information from the globus pallidus and the ventral pallidum at different levels of the basal ganglia. (4) The interconnections between the two divisions of the pallidal complex and the subthalamic nucleus and the characterization of the neuronal network underlying the indirect pathways. The findings summarized in this commentary confirm and elaborate the models of the direct and indirect pathways of information flow through the basal ganglia and provide a morphological framework for future studies. (C) 1998 IBRO. Published by Elsevier Science Ltd.
Cholinergic neurons in the monkey neostriatum were examined at the light and electron microscopic level by immunohistochemical methods in order to localize choline acetyltransferase (ChAT), the synthesizing enzyme for acetylcholine. At the light microscopic level a sparse distribution of cholinergic neurons was identified throughout the caudate nucleus. Neurons had large (25–30 μm) somata, eccentric invaginated nuclei, primary dendrites of unequal diameters, and varicosities on distal dendritic branches. Ultrastructural study showed that the cholinergic cells had a cytoplasm abundant in organelles. Within dendritic branches, mitochondria and cisternae were localized primarily to varicosities, Synaptic inputs were distributed mostly to the dendrites and at least four types that formed symmetric or asymmetric synapses were observed.
Herding behavior arises as the collective action by individual agents that base their movement only on information about their neighbors; global properties arise from strictly local interactions. We use simulation models to explore the influence of individual behavioral rules on group pattern, and to investigate the mechanisms underlying such patterns.
Single-cell labeling experiments in cynomolgus monkeys have revealed that the subthalamic nucleus (STN) harbors several subtypes of projection neurons, each endowed with a highly patterned set of axon collaterals. This organizational feature allows single STN neurons to act directly upon the two major output structures of the basal ganglia--the substantia nigra pars reticulata and the internal pallidum--and, at the same time, to exert a multifarious effect upon the external pallidum with which the STN is reciprocally connected. These findings have clarified the role of the STN in basal ganglia organization and led to the elaboration of more accurate computational models of deep brain stimulation, a therapeutic approach currently used to alleviate the motor symptoms of Parkinson's Disease.
Examination of the nestriatum of monkeys prepared by the Golgi-Kopsch perfusion method revealed the presence of at least 6 neuronal types. The spiny type I is medium size with a high density of dendritic spines. The axon extends well beyond the dendritic field and gives off many collaterals. The spiny type II is either medium or large size, has long thick dendrites with a relatively low density of spines, and an axon similar to that of the previous type but with fever collaterals. The aspiny type I is medium size with varicose dendrites and a thin axon arborizing in the immediate vicinity of the soma. The aspiny type II is large, with many thick and thin varicose dendrites. The aspiny type III is medium size with smooth dendrites and an axon ramifying profusely within the dendritic field. The neurogliform cell is small with many branching processes. Findings indicate that the neostriatum has 2 distinct types of spiny neurons with long axons (spiny I and II), some of which may contribute to the efferent system. There are also 2 (aspiny I and III) or perhaps as many as 4 categories (aspiny I, II, III and neurogliform) of typical Golgi type II cells. Large neurons belong to 2 separate populations, one with dendritic spines and a long axon (large version of spiny II), and one with varicosities and presumably a short axon (aspiny II). A realistic interpretation of neurophysiologic data on the neostriatum must take into account all cell types instead of the current view of considering it as a pool of interneurons with few output cells.
The striatum, the main component of the basal ganglia, is composed of mainly one type of neuron, the so-called medium spiny neuron. This neuron cell type, which constitutes over 90% of striatal neurons, is the major output neuron of the striatum. Combined ultrastructural neuroanatomical methods have elucidated the organization of afferent connectivity to these neurons. The major physiologic function of striatal efferent activity appears to be inhibition of tonically active GABAergic neurons in the globus pallidus and substantia nigra pars reticulata. Thus, the excitatory input from the cerebral cortex, whose afferents make asymmetric synapses with the spines of medium spiny neurons, appears to drive the efferent activity of the striatum. Other extrinsic and intrinsic afferent synapses are situated in a position to regulate the effect of the corticostriatal excitatory input to the medium spiny neurons. For example, dopaminergic afferents from the midbrain make mainly symmetric synapses with the spine necks and dendritic shafts of the medium spiny neurons. Medium spiny neurons themselves have local axon collaterals, in addition to their efferent axon that exits the striatum, which serve to link together local clusters of medium spiny neurons. These local axon collaterals, which contain either GABA, substance P, or enkephalin, also make mainly symmetric synapses with the necks of spines or dendritic shafts of medium spiny neurons. Other afferents with similar synaptic connections to these neurons arise from cholinergic or somatostatinergic striatal intrinsic neurons. Additionally, the patterns of extrinsic and intrinsic afferents to medium spiny neurons and their extrinsic projections are related to the organization of medium spiny neurons into two mosaically organized macroscopic compartments, the striatal patches and matrix.