Skip to main content


The majority of quantitative experimental data in the neurosciences is either at the microscopic, single-cell level or at the macroscopic level of the brain and organism as a whole. My research aims to bridge the gaps between these levels of description by developing the framework required to predict the emergent states and information processing possible in nervous systems from the properties of the stochastic component neurons. Recent work has concentrated on; experimental derivation and testing of reduced neuron models, methods for solving the network dynamics of reduced neuron models, methods for extracting synaptic amplitudes from experiment, models of shot-noise synaptic drive and conductance increase, short-term synaptic dynamics, and the shaping of spiking dynamics by sub-threshold resonance in neurons. Earlier research in the neurosciences involved top-down approaches that addressed planning strategies for arm movements and locomotion. Before that I worked as a theoretical physicist applying renormalization, field-theoretic and algebraic approaches to reaction-diffusion systems in confined geometries.

Firing-Rate Response of a Neuron Receiving Excitatory and Inhibitory Synaptic Shot Noise
The synaptic coupling between neurons in neocortical networks is sufficiently strong so that relatively few synchronous synaptic pulses are required to bring a neuron from rest to the spiking threshold. However, such finite-amplitude effects of fluctuating synaptic drive are missed in the standard diffusion approximation. In Richardson and Swarbrick (2010) exact solutions for the firing-rate response to modulated presynaptic rates are derived for a neuron receiving additive excitatory and inhibitory synaptic shot noise with exponential amplitude distributions. The shot-noise description of the neuronal response to synaptic dynamics is shown to be richer and qualitatively distinct from that predicted by the diffusion approximation. It is also demonstrated how the framework developed here can be generalized to multiplicative shot noise so as to better capture effects of the inhibitory reversal potential.

  • Code for the paper figures: fig 1 and fig 2 in Octave/MATLAB format. The supplementary material can be found here.

Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents
Many neurons express trans-membrane channels that are gated by intracellular calcium concentrations or changes in membrane voltage. Such gating variables have their own dynamics with activation profiles and time-constants that are non-linear. In Richardson (2009) it is demonstrated that non-linear, dynamical gating can be straightforwardly combined with the voltage dynamics of the exponential integrate-and-fire (EIF) model (Fourcaud-Trocme et al, 2003). These conductance-based Generalised EIF models (GEMS) feature a high-degree of biophysical realism - a detailed spike-onset mechanism, multiple history-dependent conductances and a realistic spike shape - while still remaining computationally tractable.

Dynamic IV curves and experimentally verified reduced neuron models
Joint work with Laurent Badel, Wulfram Gerstner, Sandrine Lefort and Carl Petersen
Reduced neuron models are an important theoretical tool for studying the properties of biological neural networks. One class of reduced spiking-neuron model, the non-linear integrate-and-fire (IF) models, take the form of a stochastic first-order differential equation for the membrane voltage with some restoring “force” that can be thought of as a projection of the ionic current onto a single current-voltage relation - an IV-curve. We have recently developed a novel method for extracting a dynamic IV curve, both in the run-up to the spike and in spike-triggered mode, yielding the optimal non-linear IF model for a particular cell. Thus far we have studied the response properties of two classes of cortical neuron; layer-5 pyramidal cells (Badel et al 2008a) and fast-spiking interneurons (Badel et al 2008b). In both cases the Exponential IF neuron provides an excellent fit to the data and a highly accurate prediction of experimentally-measured spike times.

The Threshold Integration method for solving integrate-and-fire neuron models
Integrate-and-fire (IF) models are a popular choice for reduced neuron modelling. Combined with a white-noise approximation of current-based or conductance-based synaptic drive, the models capture many aspects of the spiking dynamics seen in experiments and, more importantly, provide a basis for perturbative approaches that incorporate further biophysical details. The resulting mathematical system is a one-dimensional Fokker-Planck equation with boundary conditions at the spike threshold and reset. Until recently, closed-form solutions were only available for the linear, leaky IF model. Through the development of the Threshold Integration (ThIn) method we have recently provided a solution for the full family of non-linear IF models (Richardson, 2007; Richardson, 2008). The ThIn method is efficient and rather simple to implement.

  • Octave/MATLAB code for the steady-state rate and firing-rate modulation for the leaky and exponential IF models can be found here.

The voltage deconvolution method for measurement of synaptic amplitudes
Joint work with Gilad Silberberg
common experimental protocol for measuring short-term synaptic dynamics – the use dependent strength of synapses over timescales of 10s-100s of microseconds – is to measure the amplitudes of post-synaptic potentials (PSPs) triggered by a controlled train of pre-synaptic spikes. Amplitude measurement is complicated by the membrane filtering, causing PSPs to overlap and requiring that the effects of previous PSPs be subtracted from the voltage trace before the next PSP can be measured. We recently developed a labour-saving voltage-deconvolution method that removes the membrane filtering from the entire trace (and working also in the presence of voltage-gated currents such as Ih) thereby greatly simplifying the measurement process (Richardson and Silberberg, 2008). The technique provides amplitudes identical to the standard method of subtraction, but allows for the automation of the measurement process crucial for high-throughput measurements. Example voltage traces and the code required to analyse them are available on request and will be made available here shortly.

Synaptic conductance fluctuations and shot noise
Joint work with Wulfram Gerstner
Synapses work by opening channels in the post-synaptic neuron leading to trans-membrane-current flow and an increased leakiness, or conductance, which can significantly modulate the integrative properties of the neuron. The relative strengths of the tonic conductance increase and the conductance and current fluctuations during synaptic bombardment have recently come under some scrutiny. We were able to show (Richardson and Gerstner, 2005) for the case of sub-threshold fluctuations with coloured synaptic drive that a consistent model of conductance fluctuations also requires the shot-noise effects of the finite-amplitude of synaptic pulses to be included.

Short-term synaptic dynamics
Joint work with Ofer Melamed, Gilad Silberberg, Wulfram Gerstner and Henry Markram
The synaptic response to a pre-synaptic pulse is history dependent. Certain synapses react with decreasing strength to a barrage of closely spaced inputs, whereas others react with increasing strength. The timescales of these effects are in the range of hundreds of milliseconds and can strongly affect the dynamical response of neural networks. During population bursts in neocortical tissue different classes of neurons react with differing delays as a function of their pre-synaptic dynamics (Silberberg et al 2004, J Physiol). We developed a biophysically detailed models which included the measured synaptic dynamics and membrane-response properties and were able to demonstrate; first, that a model without short-term synaptic dynamics could not capture the observed behaviour (Melamed et al, 2005) and second, that including the correct pre-post synaptic dynamics gave us predictions that matched experiment accurately (Richardson et al 2005). These experimental and theoretical results suggest that short-term synaptic dynamics are likely to play an important role in orchestrating the timing of activity in neocortical networks.

Subthreshold resonance and firing-rate resonance
Joint work with Nicolas Brunel and Vincent Hakim
Voltage oscillations have been seen in the nervous system since the first electrophysiological measurements. Many cells throughout the peripheral and central nervous systems display a form of frequency preference known as sub-threshold resonance. These cells react strongly to oscillating synaptic drive at a particular frequency. We developed (Richardson et al 2003; Brunel et al, 2003) a Generalised Integrate and Fire model (GIF) that captures the effects of resonance by linearizing the underlying non-linear, conductance-based model and combining this with a threshold for a spike. Using a Fokker-Planck, population-based approach, we were able to demonstrate that in certain parameter ranges neurons with a sub-threshold resonance can broadcast their frequency preference to the network. This work draws a causal link between cellular and network properties and provides further evidence for the role of single-cell frequency preference in generating or reinforcing oscillations in networks of neurons.

Motor control of human arm movements and walking around curved paths
Joint works with: Tamar Flash; Hicheur, Vieilledent, Flash and Berthoz
A great many schemes have been suggested for how brains plan movement. We demonstrated that two such phenomenological proposals, with completely different underlying philosophical motivations – the Minimum-Jerk Principle which states that the brain plans movements as a whole, minimising the rate of change of acceleration, and the Two-Thirds Power Law which states that the brain plans movements locally matching velocity to curvature – have almost identical experimental predictions (Richardson and Flash, 2002). Later research further examined the relevance of these phenomenological models to locomotion by comparing the kinematics of how people draw figures (such as an ellipse or a figure-of-eight) on paper with how they walk around the same scaled up shapes (Hicheur et al, 2005). We found surprising similarities between the kinematics of the two different modes of movement – drawing and walking – suggesting common planning at some level in the brain for these distinct motor activities.

Non-equilibrium stochastic dynamics of reaction systems in confined dimensions
Joint works with: Martin Evans; John Cardy; and Yariv Kafri
The dynamics of reactant densities in chemical reactions are typically modelled by the rate-equation formalism. However, it is known that when the reactants are not well mixed fluctuations arising from their diffusive motion can render the rate-based approach wildly inaccurate. Using field-theoretical methods and exact solutions we examined diffusion-limited effects in a variety of chemical reactions including: reaction fronts in ballistic annihilation (Richardson, 1997 using a q-deformed algebra), reaction fronts in asymmetric diffusion processes (Richardson and Evans, 1997), reactions in a Sinai-disordered medium (Richardson and Cardy, 1999) and the effects of an impenetrable boundary on chemical reactions (Richardson and Kafri, 1999; Kafri and Richardson, 1999).