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Adaptation is a ubiquitous property of sensory neurons. Multisensory neurons, receiving convergent input from different sensory modalities, also likely exhibit adaptation. The responses of multisensory superior colliculus neurons have been extensively studied, but the impact of adaptation on these responses has not been examined. Multisensory neurons in the superior colliculus exhibit cross-modal enhancement, an often non-linear and non-additive increase in response when a stimulus in one modality is paired with a stimulus in a different modality. We examine the possible impact of adaptation on cross-modal enhancement within the framework of a simple model of adaptation for a neuron employing a saturating, logistic response function. We consider how adaptation to an input's mean and standard deviation affects cross-modal enhancement, and also how the statistical correlations between two different modalities influence cross-modal enhancement. We determine the optimal bimodal stimuli to present a bimodal neuron that evoke the largest changes in cross-modal enhancement under adaptation to input statistics. The model requires separate gains for each modality, unless the statistics specific to each modality have been standardised by prior adaptation in earlier, unisensory neurons. The model also predicts that increasing the correlation coefficient between two modalities reduces a multisensory neuron's overall gain.

Sensory neurons adapt to changes in the natural statistics of their environments through processes such as gain control and firing threshold adjustment. It has been argued that neurons early in sensory pathways adapt according to information-theoretic criteria, perhaps maximising their coding efficiency or information rate. Here, we draw a distinction between how a neuron's preferred operating point is determined and how its preferred operating point is maintained through adaptation. We propose that a neuron's preferred operating point can be characterised by the probability density function (PDF) of its output spike rate, and that adaptation maintains an invariant output PDF, regardless of how this output PDF is initially set. Considering a sigmoidal transfer function for simplicity, we derive simple adaptation rules for a neuron with one sensory input that permit adaptation to the lower-order statistics of the input, independent of how the preferred operating point of the neuron is set. Thus, if the preferred operating point is, in fact, set according to information-theoretic criteria, then these rules nonetheless maintain a neuron at that point. Our approach generalises from the unimodal case to the multimodal case, for a neuron with inputs from distinct sensory channels, and we briefly consider this case too.

We argue that all embodied organisms, whether robots or animals, face the same challenge: of adapting to bodies, brains and environments that undergo constant and inevitable change. After highlighting the evidence for the universal role of a class of molecular factors called neurotrophic factors in the response of animals to this challenge, we suggest that implementing models of neurotrophic interactions on robots may confer on them the adaptability and robustness exhibited by animals. We briefly review a mathematical model of neurotrophic interactions and then discuss its application in a robotic context. Finally, we examine the potential, or otherwise, of our approach to developmental robotics.

Synaptic normalization is used to enforce competitive dynamics in many models of developmental synaptic plasticity. In linear and semilinear Hebbian models, multiplicative synaptic normalization fails to segregate afferents whose activity patterns are positively correlated. To achieve this, the biologically problematic device of subtractive synaptic normalization must be used instead. Our own model of competition for neurotrophic support, which can segregate positively correlated afferents, was developed in part in an attempt to overcome these problems by removing the need for synaptic normalization altogether. However, we now show that the dynamics of our model decompose into two decoupled subspaces, with competitive dynamics being implemented in one of them through a nonlinear Hebb rule and multiplicative synaptic normalization. This normalization is "emergent" rather than imposed. We argue that these observations permit biologically plausible forms of synaptic normalization to be viewed as abstract and general descriptions of the underlying biology in certain scaleless models of synaptic plasticity.

Recent experimental data indicate that both neurotrophic factors (NTFs) and intracortical inhibitory circuitry are implicated in the development and plasticity of ocular dominance columns. We extend a neurotrophic model of developmental synaptic plasticity, which previously failed to account correctly for the differences between monocular deprivation and binocular deprivation, and show that the inclusion of lateral cortical inhibition is indeed necessary in understanding the effects of visual deprivation in the model. In particular, we argue that monocular deprivation causes a differential shift in the balance between inhibition and excitation in cortical columns, down-regulating NTFs in deprived-eye columns and up-regulating NTFs in undeprived-eye columns; during binocular deprivation, however, no such shift occurs. We thus postulate that the response to visual deprivation is at the level of the cortical circuit, while the mechanisms of afferent segregation are at the molecular or cellular level. Such a dissociation is supported by recent experimental work challenging the assumption that columnar organisation develops in an activity-dependent, competitive fashion. Our extended model also questions recent attempts to distinguish between heterosynaptic and homosynaptic models of synaptic plasticity.

Understanding the mechanisms of competitive synaptic plasticity, both anatomical and physiological, is of central importance to developmental neuroscience. Neurotrophic factors (NTFs) are implicated at almost every level of synaptic plasticity, from rapid physiological effects to slower anatomical effects, in addition to being implicated in competitive plasticity. Previously, we have built and analysed a mathematical model of anatomical synaptic plasticity based on competition for neurotrophic support. Here, we extend our work to build a combined, anatomical and physiological model. We find that, in order to understand the mechanisms of competitive physiological plasticity, we must postulate a central role for the change in expression of NTF receptors (NTFRs) on afferent synaptic terminals. Only by supposing that the expression of NTFRs is governed by NTF uptake do we find that physiological plasticity is competitive in character. We perform a fixed point analysis that establishes when afferent segregation is possible as a function of the parameters in the model, and simulate the model numerically to shed further light on its properties. A very clear prediction emerges from our model: that, as the efficacy of a terminal that is destined to be retracted due to competitive interactions reduces to zero, the NTFRs on that terminal should be down-regulated. Furthermore, our model requires that this reduction in synaptic efficacy never occurs significantly before the down-regulation in NTFRs. Such a prediction should be testable, and renders our model capable of being invalidated, in contrast to many other models of synaptic competition, which merely impose rather than seek to illuminate the quintessential feature of developmental synaptic plasticity.

The development of the retinogeniculate pathway or the geniculocortical pathway, or both, occurs either before birth or before eye opening in many species. It is widely believed that spontaneous retinal activity could drive the segregation of afferents into eye-specific laminae or columns and the refinement of initially diffuse receptive fields and the emergence of orderly, retinotopic organization. We show that a recent computational model that generates a phenomenologically accurate representation of spontaneous retinal activity can indeed drive afferent segregation and, more particularly, topographic and receptive field refinement in the retinogeniculocortical system. We use a model of anatomical synaptic plasticity based on recent data suggesting that afferents might compete for limited amounts of retrograde neurotrophic factors (NTFs). We find that afferent segregation and receptive field formation are disrupted in the presence of exogenous NTFs. We thus predict that infusion of NTFs into the lateral geniculate nucleus should disrupt normal development and that the infusion of such factors into the striate cortex should disrupt receptive field refinement in addition to the well known disruption of ocular dominance column (ODC) formation. To demonstrate that the capacity of our model of plasticity to drive normal development is not restricted just to spontaneous retinal activity, we also use a coarse representation of visually evoked activity in some simulations. We find that such simulations can exhibit the formation of ODCs followed by their disappearance, reminiscent of the New World marmoset. A decrease in interocular correlations stabilizes these ODCs. Thus we predict that divergent strabismus should render marmoset ODCs stable into adulthood.

Neurotrophic factors, particularly the neurotrophin gene family of neurotrophic factors, are implicated in activity-dependent anatomical plasticity in the visual cortex and at the neuromuscular junction. Accumulating evidence implicates neurotrophic factors as possible mediators of activity-dependent competition between afferents, leading to the segregation of afferents' arbors on the target space. We present a biologically plausible mathematical model of competition for neurotrophic factors. We show that the model leads to anatomical segregation, provided that the levels of neurotrophic factors released in an activity-independent manner, or the levels available by exogenous infusion, are below a critical value, which we derive. Above this critical value, afferent segregation breaks down. We also show that the model segregates afferents even in the presence of very highly correlated patterns of afferent activity. The model is therefore ideally suited for application to the development of ocular dominance columns in the kitten visual cortex.

Activity-dependent competition between afferents in the primary visual cortex of many mammals is a quintessential feature of neuronal development. From both experimental and theoretical perspectives, understanding the mechanisms underlying competition is a significant challenge. Recent experimental work suggests that geniculocortical afferents might compete for retrograde neurotrophic factors. We show that a mathematically well-characterized model of retrograde neurotrophic interactions, in which the afferent uptake of neurotrophic factors is activity-dependent and in which the average level of uptake determines the complexity of the axonal arbors of afferents, permits the anatomical segregation of geniculocortical afferents into ocular dominance columns. The model induces segregation provided that the levels of neurotrophic factors available either by activity-independent release from cortical cells or by exogenous cortical infusion are not too high; otherwise segregation breaks down. We show that the model exhibits changes in ocular dominance column periodicity in response to changes in interocular image correlations and that the model predicts that changes in intraocular image correlations should also affect columnar periodicity.

We construct a model of activity-dependent, anatomical inhibitory plasticity. We apply the model to the mammalian auditory system. Specifically, we model the activity-dependent topographic refinement of inhibitory projections in the auditory brain stem, and we construct an anatomically abstract model of binaural band formation in the primary auditory cortex involving the segregation of different populations of inhibitory and excitatory afferents. Issues raised and predictions made include the nature of interactions between excitatory and inhibitory afferents innervating the same population of target cells, and the possibility that pharmacological manipulations of the developing primary auditory cortex might induce a shift in the periodicity of binaural bands. Any model of inhibitory plasticity must confront the issue of postulating mechanisms underlying such plasticity. In order to attempt to understand, at least theoretically, what the mechanisms underlying inhibitory plasticity might be, we propose the existence of a new class of neurotrophic factors that promote neurite outgrowth from and mediate competitive interactions between inhibitory afferents. We suppose that such factors are up-regulated by hyperpolarisation and down-regulated by depolarisation. Furthermore, we suppose that their activity-dependent release from target cells depends on Cl- influx. Such factors are therefore assumed to be the physiological inverse of such factors as nerve growth factor and brain-derived neurotrophic factor, which are up-regulated by depolarisation and down-regulated by hyperpolarisation, with their activity-dependent release depending on Na+, and not Ca2+, influx.

We examine the claim that a class of sprouting-and-retraction models is mathematically equivalent to a fixed-anatomy model. We accept, subject to important caveats, a narrow mathematical equivalence of the energy functions in both classes of model. We argue that this narrow equivalence of energy functions does not, however, entail equivalence of the models. Indeed, the claim of complete model equivalence hides significant dynamical differences between the approaches, which we discuss. We also disagree that our work demonstrates that subtractive constraint enforcement is natural in fixed-anatomy models.

In previous work we have developed a computational framework for topographic map formation and plasticity based on axonal process sprouting and retraction, in which sprouting and retraction are governed by competition for neurotrophic support. Here we show that such an approach can account for certain aspects of the dendritic morphology of cortical maps. In particular, we model the development of ocular dominance columns in the primary visual cortex and show that cortical cells near to column boundaries prefer to elaborate dendritic fields which avoid crossing the boundaries. This emerges as different functional inputs are spatially separated. We predict that afferent segregation occurs before or simultaneously with, but not after, the emergence of dendritic bias. We predict that animals reared with complete but asynchronous stimulation of the optic nerves do not develop a dendritic bias. We suggest that the emergence of a dendritic bias might provide a partial account for the critical period for a response to monocular deprivation. In particular, we predict that animals reared with asynchronous optic nerve stimulation might exhibit an extended critical period. Our results also indicate that the number of synapses supported by cortical cells depends on the intra-ocular image correlations used in our simulations. This suggests that inter-ocular image correlations, and thus strabismic rearing of kittens, may also affect the innervation density.

Mathematical or computational models of activity-dependent neural competition typically impose competition in anatomically fixed networks by the use of synaptic normalisation, for which there is very little experimental support. Recent experimental evidence, however, strongly implicates neurotrophic factors in neural plasticity and competition, in addition to their well-known potent effects on neurite outgrowth and synaptogenesis. We therefore present a simple, mathematical model of anatomical segregation induces by activity-dependent competition for a limited supply of a neurotrophic factor provided by target cells to afferents. We extract the behaviour of the model in various regimes, in which the neurotrophic factor is either in critical supply or in abundant supply, by a combination of analytical and numerical methods, and study the effects of correlations in afferent inputs on competition. We apply the model to three different systems: ocular dominance column formation; elimination of polyneuronal innervation at the vertebrate neuromuscular junction; trigeminal brain stem whisker-related structure formation. Several classes of related predictions emerge, including the prediction that kittens reared with strabismus should require a higher concentration of neurotrophic factor infusion into their primary visual cortex than normally reared cats in order to induce the anatomical desegregation of ocular dominance columns. We also speculate on the mechanisms of support of inhibitory rather than excitatory neurons, and suggest the existence of a separate, (Cl-)-mediated activity-dependent pathway for their neurotrophic support.

We present two related computational models of ocular dominance column formation. Both address nervous system plasticity in terms of sprouting and retraction of axonal processes rather than changes in synaptic strength implied by synapse-specific Hebbian models. We employ statistical mechanics to simulate changes in the pattern of network connectivity. Our formalism uses the concept of an energy function, which we interpret as related to the levels of target-generated neurotrophins for which afferents compete. In contrast, synapse-specific Hebbian models impose synaptic normalization, for which there is little experimental evidence, in order to induce competition. Our models make many predictions which require experimental investigation. We suggest that the absence of monocular deprivation effects in the optic tectum may be due to a tendency of amphibian retinal ganglion cells to preserve the complexity of their terminal arbors. One model raises the possibility that boundaries separating columns in the mammalian cortex are poorly innervated if they have been formed by complete but asynchronous retinal activation. Both models exhibit a phase transition, suggesting a discontinuity in the transition from a binocular cortex to one possessing ocular dominance columns. Finally, our other model could account for the perpendicularity of ocular dominance columns to the boundary of the primary visual cortex while admitting of less ordered central patterns.

Following our recent presentation of an axonal process sprouting and retraction framework for ocular dominance column formation, we now apply it, unchanged, to address issues of adult somatosensory map plasticity. Specifically, we model the rearrangement of S-I in adult rodents following denervation of a row of vibrissae, and the rearrangement of area 3b in adult monkeys following hyperstimulation of a digit. While we do not attempt to capture the rapid changes which occur as the result of unmasking or potentiating existing connections, we demonstrate that axonal process sprouting and retraction is a possible mechanism mediating many of the long-term changes induced by anomalous peripheral activity. A significant feature of our framework, demonstrated by this study, is that it can account for plasticity in both developing and mature systems, and in different sensory modalities. In contrast, synapse-specific Hebbian models with synaptic normalization, which employ anatomically fixed connections capable of changes in efficacy, may not be able to account for both developmental and adult plasticity without the form of the imposed normalization, which enforces competition between afferents, being changed.

Computational models of activity-dependent competitive neural plasticity typically impose competition in networks in which plasticity is accommodated by permitting changes in the efficacies of synapses in an anatomically fixed network. This is despite the fact that much evidence suggests that neurons compete for neurotrophins, during both target innervation and activity-dependent synaptic re-arrangement, which influence the sprouting and retraction of axonal processes. We therefore present a new approach to the computational modelling of competitive neural plasticity which permits neurons to compete explicitly for neurotrophins. This competition is associated with the sprouting and retraction of axonal processes. Because there is much uncertainty regarding the basic mechanisms, we adopt the powerful machinery of statistical mechanics to avoid the need to address these issues. We show that such an approach can readily account for a wide range of plasticity phenomena in a range of systems, including the results of various pharmacological manipulations.