Elsevier

Neural Networks

Volume 21, Issues 2–3, March–April 2008, Pages 130-139
Neural Networks

2008 Special Issue
Modeling propagation delays in the development of SOMs — a parallel with abnormal brain growth in autism

https://doi.org/10.1016/j.neunet.2007.12.006Get rights and content

Abstract

Brain overgrowth in early developmental stages of children with autism is well documented. This paper explores the possibility that increases in propagation delays of stimuli and the signals triggered by them, resulting from this overgrowth, may be conducive to the development of poorly structured cortical maps, which may in turn be associated with autistic characteristics. We use a framework based on Self-Organizing Maps (SOMs). Unlike the conventional SOM model that assumes that all neurons in the neighborhood of the neuron closest to a stimulus instantaneously react to it and adjust their weights, we propose a more biologically realistic model that acknowledges delays inherent in the propagation of signals. We show that propagation delays can significantly affect the performance of SOMs. Coverage of stimuli is negatively affected by either an increase in the dilution factor (a parameter in the proposed model that controls the adjustment of responses to overlapping stimuli), or a decrease in propagation speed. For large dilution factors the topological structure of the maps is also compromised. We also demonstrate the model’s robustness to different input stimuli layouts and distributions.

Introduction

Autism is a pervasive developmental disorder characterized by three core areas of impairment: socialization, communication and imagination. Although the precise mechanisms that cause it have not yet been established, it is now well accepted that its origins are in neurological dysfunctions (Frith, 2003, Volkmar et al., 2005). It is best characterized as a spectrum of disorders that vary in the severity of symptoms and age of onset. Manifestations also vary considerably within an individual over time, but generally, autism has lifelong effects on learning, self-care and the ability to participate in the community.

Ample evidence of abnormal brain development in autism has been provided by neuroanatomical studies (Bauman & Kemper, 2005). Differences have been reported in the cerebellum, portions of the limbic system, and the cerebral cortex. The most consistent pathological finding is the marked decrease of Purkinje cells in the cerebellum (Bauman & Kemper, 1996).

Recent MRI studies by Courchesne et al. (Courchesne et al., 2001, Courchesne et al., 1999) have demonstrated abnormal regulation of brain growth in autism, with overgrowth in early life followed by abnormally slow growth in some regions, and premature cessation of growth in others. At birth, brain size appears to be normal, but by 2–4 years of age brain volumes are larger than average in about 90% of children. By adolescence brain volumes appear to return to normal. The MRI studies by Aylward, Minshew, Field, Sparks, and Singh (2002) have also shown accelerated brain growth in early life resulting in brain enlargement in childhood. The authors conclude that normal brain volume in adolescents and adults is due to a slight decrease in volume for these individuals at the same time as normal children are experiencing a slight increase. Although numerous neurobiological mechanisms have been proposed to explain the apparent early brain overgrowth in autism–increased neurogenesis, decreased neuronal cell death, increased production of non-neuronal brain tissues, decreased synaptic pruning, and abnormalities of myelin–there is no firm pathological evidence to support any of them (Bauman & Kemper, 2005).

This paper elaborates on earlier work by Noriega (2007a). It explores the possibility that increases in propagation delays of stimuli and the signals triggered by them, resulting from brain overgrowth in early stages of development in autistic children, may be conducive to the development of poorly structured cortical maps, which may in turn be associated with autistic characteristics.

We introduce a biologically plausible model of cortical areas of the brain that takes into consideration delays inherent in the propagation of signals. The model is based on Self-Organizing Maps (SOMs). These are artificial neural networks in which a structure of neurons, upon training, becomes sensitive to features in the stimuli. SOMs, originally developed by Kohonen (1997), have been in use for more than two decades, extensively applied in fields ranging from engineering to medicine, biology and economics. More than 5000 articles are cited by Oja, Kaski, and Kohonen (2002).

Our model is effectively an enhanced-SOM, which accounts for propagation delays. This is in contrast to the conventional SOM algorithm which assumes that all the neurons in the neighborhood of the neuron closest to a stimulus instantaneously react to it. In modeling the response to overlapping stimuli that result from propagation delays, we introduce the concept of a dilution factor that modulates the adjustment of neuron weights.

In a first attempt to study the brain overgrowth characteristic of autism using SOMs (Noriega, 2007b), the dimensions of the network were dynamically adjusted to simulate growth abnormalities.

Several computational models addressing various aspects of acquired and developmental disorders have been proposed in the past. In the area of autism, a number of contributions have been built within the framework of SOMs (de Carvalho et al., 1999, Gustafsson, 1997, Gustafsson and Paplinski, 2002, Gustafsson and Paplinski, 2004, Noriega, 2007b, Paplinski and Gustafsson, 2004) while some have been based on other types of connectionist networks (Balkenius and Bjorne, 2004, Cohen, 1994, Cohen, 1998, McClelland, 2000, O’Loughlin and Thagard, 2000). SOMs have also been used to model other forms of cognitive and developmental disorders (Spitzer, 1995), and to suggest models that, while adhering to constraints of biological plausibility, do not suggest the manner in which the disordered SOMs may affect the functioning of the system (Oliver, Johnson, & Pennington, 2000). A review of models of brain and cognitive disorders, some utilizing the SOM framework, is offered by Reggia, Ruppin, and Berndt (1996). Perhaps the most comprehensive attempt at modeling autism is the recent work by Grossberg and Seidman (2006). They introduce a neural model which proposes how cognitive, emotional, timing, and motor processes may interact together to create and perpetuate autistic symptoms. The model shows how autistic behavioral symptoms may arise from prescribed breakdowns in these brain processes that involve brain regions like the prefrontal and temporal cortex, amygdala, hippocampus, and cerebellum.

A number of models have been developed that integrate temporal information into the SOM. Rather than addressing the issue of propagation delays within the network, they are designed for dynamic sensory processing problems–see Euliano and Principe (1999) and the references therein.

Section snippets

Biological context

Fundamentally, the biological analog of the SOM is that of cortical areas consisting of several layers of neurons organized into columnar structures. In the visual cortex, for example, cooperation and competition between axons influence the development of ocular dominance columns primarily during a critical period of development (Kandel, Schwartz, & Jessel, 2000). The balance of activity in the fibers from the two eyes affects the segregation of afferent fibers and the establishment of the

Overview and terminology

We start by presenting a brief description of SOMs to clarify notation–see, for example, Kohonen (1997) or Ritter, Martinetz, and Schulten (1992) for details. SOMs are competitive neural networks that perform a mapping of a p-dimensional input space into an l-dimensional feature space. At time (or iteration) k, each neuron i is characterized by the weight vector wi(k)=[wi1(k)wi2(k)wip(k)], and a position in an l-dimensional lattice given by the vector vi(k)=[vi1(k)vi2(k)vil(k)]. Each input

Ring-shaped area with uniform stimuli distribution

In a first set of simulations 1000 stimuli are uniformly distributed into a ring-shaped area in a 2-dimensional Cartesian plane. The ring is centered at (0.5,0.5) and has exterior radius re=0.35 and interior radius ri=0.15. In each epoch of the training process the stimuli are presented to the network in random order (see Fig. 2 for reference). The choice of a ring-shaped region demands precise positioning of the neurons to achieve a good measure of coverage (see below), which helps emphasize

Discussion

The work presented is motivated by attempts to model brain mechanisms that may be associated with autism. Brain overgrowth in early developmental stages of children with autism is now well documented. The proposed SOM-based model of cortical areas of the brain is biologically plausible, and suggests mechanisms by which increases in propagation delays resulting from this overgrowth may lead to poorly structured cortical feature maps.

Cortical maps with well-structured topologies have important

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    An abbreviated version of some portions of this article appeared in Noriega (2007a) as part of the IJCNN 2007 Conference Proceedings, published under IEE copyright.

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