Modelling Feedback Excitation, Pacemaker Properties and Sensory Switching of Electrically Coupled Brainstem Neurons Controlling Rhythmic Activity.

What cellular and network properties allow reliable neuronal rhythm generation or firing that can be started and stopped by brief synaptic inputs? We investigate rhythmic activity in an electrically-coupled population of brainstem neurons driving swimming locomotion in young frog tadpoles, and how activity is switched on and off by brief sensory stimulation. We build a computational model of 30 electrically-coupled conditional pacemaker neurons on one side of the tadpole hindbrain and spinal cord. Based on experimental estimates for neuron properties, population sizes, synapse strengths and connections, we show that: long-lasting, mutual, glutamatergic excitation between the neurons allows the network to sustain rhythmic pacemaker firing at swimming frequencies following brief synaptic excitation; activity persists but rhythm breaks down without electrical coupling; NMDA voltage-dependency doubles the range of synaptic feedback strengths generating sustained rhythm. The network can be switched on and off at short latency by brief synaptic excitation and inhibition. We demonstrate that a population of generic Hodgkin-Huxley type neurons coupled by glutamatergic excitatory feedback can generate sustained asynchronous firing switched on and off synaptically. We conclude that networks of neurons with NMDAR mediated feedback excitation can generate self-sustained activity following brief synaptic excitation. The frequency of activity is limited by the kinetics of the neuron membrane channels and can be stopped by brief inhibitory input. Network activity can be rhythmic at lower frequencies if the neurons are electrically coupled. Our key finding is that excitatory synaptic feedback within a population of neurons can produce switchable, stable, sustained firing without synaptic inhibition.

Modelling the Effects of Electrical Coupling between Unmyelinated Axons of Brainstem Neurons Controlling Rhythmic Activity.

Gap junctions between fine unmyelinated axons can electrically couple groups of brain neurons to synchronise firing and contribute to rhythmic activity. To explore the distribution and significance of electrical coupling, we modelled a well analysed, small population of brainstem neurons which drive swimming in young frog tadpoles. A passive network of 30 multicompartmental neurons with unmyelinated axons was used to infer that: axon-axon gap junctions close to the soma gave the best match to experimentally measured coupling coefficients; axon diameter had a strong influence on coupling; most neurons were coupled indirectly via the axons of other neurons. When active channels were added, gap junctions could make action potential propagation along the thin axons unreliable. Increased sodium and decreased potassium channel densities in the initial axon segment improved action potential propagation. Modelling suggested that the single spike firing to step current injection observed in whole-cell recordings is not a cellular property but a dynamic consequence of shunting resulting from electrical coupling. Without electrical coupling, firing of the population during depolarising current was unsynchronised; with coupling, the population showed synchronous recruitment and rhythmic firing. When activated instead by increasing levels of modelled sensory pathway input, the population without electrical coupling was recruited incrementally to unpatterned activity. However, when coupled, the population was recruited all-or-none at threshold into a rhythmic swimming pattern: the tadpole "decided" to swim. Modelling emphasises uncertainties about fine unmyelinated axon physiology but, when informed by biological data, makes general predictions about gap junctions: locations close to the soma; relatively small numbers; many indirect connections between neurons; cause of action potential propagation failure in fine axons; misleading alteration of intrinsic firing properties. Modelling also indicates that electrical coupling within a population can synchronize recruitment of neurons and their pacemaker firing during rhythmic activity.

Analysis of local and global topographic order in mouse retinocollicular maps.

We introduce the Lattice Method for the quantitative assessment of the topographic order within the pattern of connections between two structures. We apply this method to published visuocollicular mapping data obtained by Fourier-based intrinsic imaging of mouse colliculus. We find that, in maps from wild types and ß2 knock-outs, at least 150 points on the colliculus are represented in the visual field in the correct relative order. In maps from animals with knock-out of the three ephrinA ligands (TKO), thought to specify the rostrocaudal axis of the map, the projection on the colliculus of each small circular area of visual field is elongated approximately rostrocaudally. Of these projections, 9% are made up of two distinct regions lying along the direction of ingrowth of retinal fibers. These are similar to the ectopic projections found in other ephrinA knock-out data. Coexisting with the ectopic projections, each TKO map contains a submap where neighbor-neighbor relations are preserved, which is ordered along both rostrocaudal and mediolateral axes, in the orientation found in wild-type maps. The submaps vary in size with order well above chance level, which can approach the order in wild-type maps. Knock-out of both ß2 and two of the three ephrinAs yields maps with some order. The ordered TKO maps cannot be produced by correlated neural activity acting alone, as this mechanism is unable to specify map orientation. These results invite reassessment of the role of molecular signaling, particularly that of ephrinAs, in the formation of ordered nerve connections.

Standard anatomical and visual space for the mouse retina: computational reconstruction and transformation of flattened retinae with the Retistruct package.

The concept of topographic mapping is central to the understanding of the visual system at many levels, from the developmental to the computational. It is important to be able to relate different coordinate systems, e.g. maps of the visual field and maps of the retina. Retinal maps are frequently based on flat-mount preparations. These use dissection and relaxing cuts to render the quasi-spherical retina into a 2D preparation. The variable nature of relaxing cuts and associated tears limits quantitative cross-animal comparisons. We present an algorithm, "Retistruct," that reconstructs retinal flat-mounts by mapping them into a standard, spherical retinal space. This is achieved by: stitching the marked-up cuts of the flat-mount outline; dividing the stitched outline into a mesh whose vertices then are mapped onto a curtailed sphere; and finally moving the vertices so as to minimise a physically-inspired deformation energy function. Our validation studies indicate that the algorithm can estimate the position of a point on the intact adult retina to within 8° of arc (3.6% of nasotemporal axis). The coordinates in reconstructed retinae can be transformed to visuotopic coordinates. Retistruct is used to investigate the organisation of the adult mouse visual system. We orient the retina relative to the nictitating membrane and compare this to eye muscle insertions. To align the retinotopic and visuotopic coordinate systems in the mouse, we utilised the geometry of binocular vision. In standard retinal space, the composite decussation line for the uncrossed retinal projection is located 64° away from the retinal pole. Projecting anatomically defined uncrossed retinal projections into visual space gives binocular congruence if the optical axis of the mouse eye is oriented at 64° azimuth and 22° elevation, in concordance with previous results. Moreover, using these coordinates, the dorsoventral boundary for S-opsin expressing cones closely matches the horizontal meridian.

Similarity-based extraction of individual networks from gray matter MRI scans.

The characterization of gray matter morphology of individual brains is an important issue in neuroscience. Graph theory has been used to describe cortical morphology, with networks based on covariation of gray matter volume or thickness between cortical areas across people. Here, we extend this research by proposing a new method that describes the gray matter morphology of an individual cortex as a network. In these large-scale morphological networks, nodes represent small cortical regions, and edges connect regions that have a statistically similar structure. The method was applied to a healthy sample (n = 14, scanned at 2 different time points). For all networks, we described the spatial degree distribution, average minimum path length, average clustering coefficient, small world property, and betweenness centrality (BC). Finally, we studied the reproducibility of all these properties. The networks showed more clustering than random networks and a similar minimum path length, indicating that they were "small world." The spatial degree and BC distributions corresponded closely to those from group-derived networks. All network property values were reproducible over the 2 time points examined. Our results demonstrate that intracortical similarities can be used to provide a robust statistical description of individual gray matter morphology.

Reanalysis of EphA3 Knock-In Double Maps in Mouse Suggests That Stochasticity in Topographic Map Formation Acts at the Retina Rather than between Competing Mechanisms at the Colliculus.

It has been suggested that stochasticity acts in the formation of topographically ordered maps in the visual system through the opposing chemoaffinity and neural activity forces acting on the innervating nerve fibers being held in an unstable equilibrium. Evidence comes from the Islet2-EphA3 knock-in mouse, in which ∼50% of the retinal ganglion cells, distributed across the retina, acquire the EphA3 receptor, thus having an enhanced density of EphA which specifies retinotopic order along the rostrocaudal (RC) axis of the colliculus. Sampling EphA3 knock-in maps in heterozygotes at different positions along the mediolateral (ML) extent of the colliculus had found single 1D maps [as in wild types (WTs)], double maps (as in homozygous knock-ins) or both single and double maps. We constructed full 2D maps from the same mouse dataset. We found either single maps or maps where the visual field projects rostrally, with a part-projection more caudally to form a double map, the extent and location of this duplication varying considerably. Contrary to previous analyses, there was no strict demarcation between heterozygous and homozygous maps. These maps were replicated in a computational model where, as the level of EphA3 was increased, there was a smooth transition from single to double maps. Our results suggest that the diversity in these retinotopic maps has its origin in a variability over the retina in the effective amount of EphA3, such as through variability in gene expression or the proportion of EphA3+ retinal ganglion cells, rather than the result of competing mechanisms acting at the colliculus.

Optimal learning rules for familiarity detection.

It has been suggested that the mammalian memory system has both familiarity and recollection components. Recently, a high-capacity network to store familiarity has been proposed. Here we derive analytically the optimal learning rule for such a familiarity memory using a signal- to-noise ratio analysis. We find that in the limit of large networks the covariance rule, known to be the optimal local, linear learning rule for pattern association, is also the optimal learning rule for familiarity discrimination. In the limit of large networks, the capacity is independent of the sparseness of the patterns and the corresponding information capacity is 0.057 bits per synapse, which is somewhat less than typically found for associative networks.

Quantitative assessment of computational models for retinotopic map formation.

Molecular and activity-based cues acting together are thought to guide retinal axons to their terminal sites in vertebrate optic tectum or superior colliculus (SC) to form an ordered map of connections. The details of mechanisms involved, and the degree to which they might interact, are still not well understood. We have developed a framework within which existing computational models can be assessed in an unbiased and quantitative manner against a set of experimental data curated from the mouse retinocollicular system. Our framework facilitates comparison between models, testing new models against known phenotypes and simulating new phenotypes in existing models. We have used this framework to assess four representative models that combine Eph/ephrin gradients and/or activity-based mechanisms and competition. Two of the models were updated from their original form to fit into our framework. The models were tested against five different phenotypes: wild type, Isl2-EphA3(ki/ki), Isl2-EphA3(ki/+), ephrin-A2,A3,A5 triple knock-out (TKO), and Math5(-/-) (Atoh7). Two models successfully reproduced the extent of the Math5(-/-) anteromedial projection, but only one of those could account for the collapse point in Isl2-EphA3(ki/+). The models needed a weak anteroposterior gradient in the SC to reproduce the residual order in the ephrin-A2,A3,A5 TKO phenotype, suggesting either an incomplete knock-out or the presence of another guidance molecule. Our article demonstrates the importance of testing retinotopic models against as full a range of phenotypes as possible, and we have made available MATLAB software, we wrote to facilitate this process.

morphforge: a toolbox for simulating small networks of biologically detailed neurons in Python.

The broad structure of a modeling study can often be explained over a cup of coffee, but converting this high-level conceptual idea into graphs of the final simulation results may require many weeks of sitting at a computer. Although models themselves can be complex, often many mental resources are wasted working around complexities of the software ecosystem such as fighting to manage files, interfacing between tools and data formats, finding mistakes in code or working out the units of variables. morphforge is a high-level, Python toolbox for building and managing simulations of small populations of multicompartmental biophysical model neurons. An entire in silico experiment, including the definition of neuronal morphologies, channel descriptions, stimuli, visualization and analysis of results can be written within a single short Python script using high-level objects. Multiple independent simulations can be created and run from a single script, allowing parameter spaces to be investigated. Consideration has been given to the reuse of both algorithmic and parameterizable components to allow both specific and stochastic parameter variations. Some other features of the toolbox include: the automatic generation of human-readable documentation (e.g., PDF files) about a simulation; the transparent handling of different biophysical units; a novel mechanism for plotting simulation results based on a system of tags; and an architecture that supports both the use of established formats for defining channels and synapses (e.g., MODL files), and the possibility to support other libraries and standards easily. We hope that this toolbox will allow scientists to quickly build simulations of multicompartmental model neurons for research and serve as a platform for further tool development.

Prevalence and elimination of sibling neurite convergence in motor units supplying neonatal and adult mouse skeletal muscle.

During development, neurons form supernumerary synapses, most of which are selectively pruned leading to stereotyped patterns of innervation. During the development of skeletal muscle innervation, or its regeneration after nerve injury, each muscle fiber is transiently innervated by multiple motor axon branches but eventually by a single branch. The selective elimination of all but one branch is the result of competition between the converging arbors. It is thought that motor neurons initially innervate muscle fibers randomly, but that axon branches from the same neuron (sibling branches) do not converge to innervate the same muscle fiber. However, random innervation would result in many neonatal endplates that are co-innervated by sibling branches. To investigate whether this occurs we examined neonatal levator auris longus (LAL) and 4th deep lumbrical (4DL) muscles, as well as adult reinnervated deep lumbrical muscles (1-4) in transgenic mice expressing yellow fluorescent protein (YFP) as a reporter. We provide direct evidence of convergence of sibling neurites within single fluorescent motor units, both during development and during regeneration after nerve crush. The incidence of sibling neurite convergence was 40% lower in regeneration and at least 75% lower during development than expected by chance. Therefore, there must be a mechanism that decreases the probability of its occurrence. As sibling neurite convergence is not seen in normal adults, or at later timepoints in regeneration, synapse elimination must also remove convergent synaptic inputs derived from the same motor neuron. Mechanistic theories of synaptic competition should now accommodate this form of isoaxonal plasticity.

Spike-timing-dependent plasticity with weight dependence evoked from physical constraints.

Analogue and mixed-signal VLSI implementations of Spike-Timing-Dependent Plasticity (STDP) are reviewed. A circuit is presented with a compact implementation of STDP suitable for parallel integration in large synaptic arrays. In contrast to previously published circuits, it uses the limitations of the silicon substrate to achieve various forms and degrees of weight dependence of STDP. It also uses reverse-biased transistors to reduce leakage from a capacitance representing weight. Chip results are presented showing: various ways in which the learning rule may be shaped; how synaptic weights may retain some indication of their learned values over periods of minutes; and how distributions of weights for synapses convergent on single neurons may shift between more or less extreme bimodality according to the strength of correlational cues in their inputs.

Synaptic rewiring for topographic mapping and receptive field development.

A model of topographic map refinement is presented which combines both weight plasticity and the formation and elimination of synapses, as well as both activity-dependent and activity-independent processes. The question of whether an activity-dependent process can refine a mapping created by an activity-independent process is addressed statistically. A new method of evaluating the quality of topographic projections is presented which allows independent consideration of the development of the centres and spatial variances of receptive fields for a projection. Synapse formation and elimination embed in the network topology changes in the weight distributions of synapses due to the activity-dependent learning rule used (spike-timing-dependent plasticity). In this model, the spatial variance of receptive fields can be reduced by an activity-dependent mechanism with or without spatially correlated inputs, but the accuracy of receptive field centres will not necessarily improve when synapses are formed based on distributions with on-average perfect topography.

Large developing receptive fields using a distributed and locally reprogrammable address-event receiver.

A distributed and locally reprogrammable address-event receiver has been designed, in which incoming address-events are monitored simultaneously by all synapses, allowing for arbitrarily large axonal fan-out without reducing channel capacity. Synapses can change the address of their presynaptic neuron, allowing the distributed implementation of a biologically realistic learning rule, with both synapse formation and elimination (synaptic rewiring). Probabilistic synapse formation leads to topographic map development, made possible by a cross-chip current-mode calculation of Euclidean distance. As well as synaptic plasticity in rewiring, synapses change weights using a competitive Hebbian learning rule (spike-timing-dependent plasticity). The weight plasticity allows receptive fields to be modified based on spatio-temporal correlations in the inputs, and the rewiring plasticity allows these modifications to become embedded in the network topology.

Asynchronous inputs and NMDA conductances predict excitatory responses in the cortical-cA1 pathway of the hippocampus.

In the hippocampus, CA1 place cells are driven by a substantial input from CA3. There is a second pathway to CA1 from the entorhinal cortex. The mode of action of cortex on CA1 through this pathway is not known. The pathway supports CA1 place field activity after CA3 has been lesioned, yet stimulation of the pathway in rat slices results in strong feedforward inhibition that prevents pyramidal cell action potentials. We use a detailed conductance-based model of this pathway to simulate the response to cortical stimulation in slice experiments and in vivo spatial exploration. We find that the presence of NMDA conductances enable CA1 pyramidal cells to integrate cortical inputs over a time scale longer than that which is effective in recruiting the inhibitory response that can suppress action potentials. We then show that this asynchronous response mode supports place field formation in response to experimentally constrained spatially modulated cortical activity. Within this model, the inclusion of GABAB conductances and the hyperpolarisation activated current I(h) reduces the strength of the GABAA inputs required to balance the excitatory inputs, and this facilitates place field formation by reducing variability in the inhibitory inputs.

The cingulate as a catalyst region for global dysfunction: a dynamical modelling paradigm.

The anterior cingulate (AC) often exhibits both structural and functional abnormalities in affective disorders. Neither the cause for this vulnerability nor its effect on behaviour is known. Due to its extensive connectivity, minor output changes from the AC may exert widespread consequences. A causal model describing coupling coefficients (effective connectivity) among several brain regions in healthy subjects performing a memory task inspired our work. This stationary causal analysis provides a theoretical framework for our nonlinear dynamical models. We tested the effects of global and local perturbations upon stability of a systems-level neural network of interconnected brain regions. Interactions between regions, represented by path coefficients, were modelled using connectivity matrices. We found that both characteristic behaviour and response to perturbation differed in networks representing perceptual matching and long-delay conditions. Owing to the highly interconnected character of the networks, activation of a few areas was sufficient to trigger characteristic patterns of behaviour. However, only perturbation of key regions resulted in global dysfunction. Likewise, recovery of function was possible by increasing output from some, but not all, regions. We suggest for this recovery to be context specific, conditional on the task, integrity of other regions and global properties such as neuronal excitability.

Influence of cell fate mechanisms upon retinal mosaic formation: a modelling study.

Many types of retinal neurone are arranged in a spatially regular manner so that the visual scene is uniformly sampled. Several mechanisms are thought to be involved in the development of regular cellular positioning. One early-acting mechanism is the lateral inhibition of neighbouring cells from acquiring the same fate, mediated by Delta-Notch signalling. We have used computer modelling to test whether lateral inhibition might transform an initial population of undifferentiated cells into more regular populations of two types of differentiated cells. Initial undifferentiated cells were positioned randomly, subject only to a minimal distance constraint. Each undifferentiated cell then acquired either primary or secondary fate using one of several lateral inhibition mechanisms. Mosaic regularity was assessed using the regularity index and the packing factor. We found that for irregular undifferentiated mosaics, the arrangement of resulting primary (but not secondary) fate cells was more regular than in the initial undifferentiated population. However, for regular undifferentiated mosaics, no further increases in the regularity of the primary fate mosaics were observed. We have used this model to test the specific hypothesis that on- and off-centre retinal ganglion cells emerge from an initial, irregular undifferentiated population of ganglion cells. Lateral inhibition can subdivide an initially irregular population into two types of cell that are mildly regular. However, lateral inhibition alone is insufficient to produce mosaics of the same regularity as observed experimentally. Likewise, and in contrast to earlier reports, cell death alone is insufficient to match the regularity of experimental mosaics. We conclude that lateral inhibition can transform irregular distributions into regular mosaics, upon which subsequent processes (such as lateral cell movement or cell death) can further refine mosaic regularity.