Multiscale modelling of biochemical networks within compartmental models

In collaboration with Professor J Douglas Armstrong and Dr Oksana Sorokina (Edinburgh).

Multiscale simulation of synaptic molecules produced with KappaNEURON.

Synaptic plasticity depends on the interaction between electrical activity in neurons and the synaptic proteome, the collection of over 1000 proteins in the post-synaptic density (PSD) of synapses. To construct models of synaptic plasticity with realistic numbers of proteins, we aim to combine rule-based models of molecular interactions in the synaptic proteome with compartmental models of the electrical activity of neurons. Rule-based models allow interactions between the combinatorially large number of protein complexes in the postsynaptic proteome to be expressed straightforwardly. Simulations of rule-based models are stochastic and thus can deal with the small copy numbers of proteins and complexes in the PSD. Compartmental models of neurons are expressed as systems of coupled ordinary differential equations and solved deterministically. We present an algorithm which incorporates stochastic rule-based models into deterministic compartmental models and demonstrate an implementation (KappaNEURON) of this hybrid system using the SpatialKappa and NEURON simulators.

Development of the nervous system

Modelling the development of neural topographic maps

In collaboration with Professor David Willshaw (Edinburgh) and Drs Stephen Eglen and Johannes Hjorth (Cambridge).

Schematic diagram of retinotopy.

During early development in vertebrates, topographic maps form between retinal ganglion cells and their targets, the optic tectum/superior colliculus and the lateral geniculate nucleus. Chemical markers, expressed in gradients, and electrical activity have been shown to influence the growth and pruning of connections between retinal ganglion cells and target cells. Chemical markers, such as Ephs and ephrins, are expressed in the membranes of retinal ganglion cell axons at levels that depend on the location of the cell body in the retina. Complementary gradients of Ephs and ephrins are expressed in gradients throughout the target region, and repulsive interactions between Ephs and ephrins are thought to inhibit axonal branching in particular regions of the target region, leading to a diffuse topographic map. Electrical activity combined with synaptic plasticity is thought to refine this mapping.

An important issue is how to quantify and analyse topographic maps. In Willshaw, Sterratt and Teriakidis (2014), we have devised a new computational method entitled the “lattice method” to analyse experimental data. Our method reveals that there is hidden order in some maps that were previously thought to be disordered.

There are a number of existing computational models of the establishment of the mapping between the retina and its targets, which have shown how both marker-based and activity-based mechanisms can set up topographic maps. In the years since these models were devised a great deal of experimental evidence has accumulated and it is important to assess whether existing models can account for new experimental data as we do in our review (Hjorth et al., 2015). The key challenge is to integrate as many of the biological constraints as possible into models that can still explain the large-scale topographic organisation of the connections.

Sterratt (2013) uses an existing model to examine how crucial it is for countergradients of ephrins in the retina and Ephs in the superior colliculus to be tuned to each other, in the presence and absence of a compensatory mechanism. Sterratt and Hjorth (2013) provide a critical commentary on a Grimbert and Cang’s (2012) model of the development of retinotopy.

Retistruct

Reconstructing a retina in Retistruct

In collaboration with Ian Thompson and Daniel Lyngholm.

We have developed Retistruct, a program to morph a flat surface with incisions (a dissected retina) onto a curvilinear surface (the original retinal shape). For more details and to download the software, see the paper or go to the Retistruct home page.

Related work is IntactEye (Hjorth et al., 2015), a software package that uses two orthogonal images of the intact retina to locate focal injections of a dye.

Learning and memory

Distance-dependent synaptic plasticity

Signals from spines in a model CA1 neuron

In hippocampal CA1 cells, synapses that are further from the cell body are larger than those closer to the cell body. In collaboration with Dr Arjen van Ooyen (Sterratt & van Ooyen 2002, 2004; Sterratt, Groen, Meredith & van Ooyen, 2012), I have been investigating whether this distance-dependent scaling could arise as a result of voltage and calcium signals elicited by synaptic inputs to the dendrites of hippocampal CA1 cells. To achieve this, we used the NEURON simulation package to implement a detailed compartmental model of a CA1 cell incorporating spines into which calcium can flow via NMDA receptors and voltage-dependent calcium channels.

In contrast to earlier modelling work, we fed more naturalistic patterns of synaptic activity into the model and our results suggest that the magnitude of calcium signals could provide the information needed for synapses to be scaled. In a separate strand of work (Sterratt & Willshaw, 2008), I have analysed the expected improvement in the performance of a CA1 cell as an associative memory due to scaling synapses appropriately for distance.

Learning and forgetting in associative memories

Results of a model of forgetting in which the rate at which synapses are weakened can be adjusted. Top The Signal to Noise Ratio (SNR) is higher for better recall, and decreases with the age of the memory. How the SNR decays with the age of a memory depends on whether synapses are weakened quickly, slowly or optimally. The point at which performance becomes unacceptable (indicated by the black line) depends on rate. Too fast, and memories disappear quickly. Too slow, and traces of old memories start to interfere with recall. Bottom Equivalently, this can be viewed as the fraction of bits that are wrong.

In collaboration with Professor David Willshaw (Sterratt & Willshaw, 2008), I have undertaken rigorous mathematical analysis and simulation of feedforward associative memories that incorporate a family of synaptic learning rules and synaptic weight decay. The networks learn newly presented memories and forget old ones. The mathematical formulation of these networks also allows the effects of distant-dependent attenuation and stochastic transmission to be modelled. For an overview of the method, see these slides from Computational Neuroscience 2006.

In collaboration with Dr Jesus Cortes (Granada) and Dr Mark van Rossum (Edinburgh), I am using mean-field analysis and simulation to investigate the performance of recurrent associative networks with sparsely-coded memory patterns, where activity levels are controlled by a mixture of inhibition and thresholds.

Familiarity memory

Familiarity memory the type of memory involved in being able to identify that a stimulus (e.g. someone met in the street) is familiar, but not being able to recall any more facts about the stimulus (e.g. the person’s name, where we know them from). Humans have a tremendous capacity for this, and in a neural network model, the number of stimuli that can be identified as familiar scales with the square of the number of neurons. This capacity is much greater than for being able to recall memories, which scales with the number of units. In collaboration with Dr Andrea Greve and Dr Mark van Rossum, I have investigated optimal learning rules for familiarity detection (Greve, Sterratt, Donaldson, Willshaw & van Rossum, 2009). Some work in preparation suggests that with a more realistic network, the capacity isn’t so great - but still better than for recall.