Dependence of the spike-triggered average voltage on membrane response properties
Introduction
Reverse-correlation methods are a standard tool in the neurosciences [2]. The spike-triggered average voltage (STV) is a quantity that contains information about both the statistics of the synaptic drive and the response properties of the neuron, and can be readily extracted from intracellular voltage traces. An analytical understanding of the processes that shape the STV can be used to classify the response characteristics of neurons and the synaptic drive to which they are subjected [1].
Typically, the neuronal models that have been used to investigate the STV are of the integrate-and-fire (IF) type with leaky, passive dynamics [1], [8]. The IF model, however, misses many important features of biological neurons such as the h-current sag seen in cortical and hippocampal pyramidal cells, as well as more complex behaviour such as damped oscillations seen in certain entorhinal cortical cells [4].
Here the form of the spike-triggered average will be investigated for model neurons with biologically more realistic dynamics. Two methods will be used: the first involves a low-noise approximation, which, in the context of spiking neurons, corresponds to low-noise, noise-driven spike generation; the second method captures the power-law behaviour of the STV near threshold—a feature missed by low-noise approximations.
Section snippets
The model
The subthreshold voltage dynamics are described by a two-variable IF model [3], [5], [9], [10] which is sufficient to capture the response of neurons with passive dynamics, h-current sag response and damped oscillations. The model comprises a variable for the membrane voltage, with time-scale and a second variable proportional to the excess current flowing through voltage-gated trans-membrane channels, with time-scale and coupled with strength to the voltage. Thus,
Low-noise approximation
Our calculation is based on a principle which states that for a set of equations of the type (1), the most likely trajectory followed by the system when moving from one point to another is obtained by maximising the probability of the synaptic input that would yield such a transition. This optimisation is carried through the minimisation of a cost function that will now be derived (the informal derivation given here follows the well-known principle of minimum available noise energy [7]).
Discussion
We have derived analytical expressions for the spike-triggered average voltage (STV) of three model neurons with subthreshold dynamics corresponding to those seen in biological neurons: passive decay, h-current sag and damped oscillations. Our analyses demonstrate that (i) the form of the STV is closely related to the subthreshold membrane dynamics, and (ii) for a hard-threshold model with white noise the STV behaves as a power-law near threshold. The distinct forms for the STV derived here
Acknowledgements
We thank Liam Paninski for useful discussions. This work was supported by Grant no. 200020-108093/1 of the Swiss National Science Foundation.
Wulfram Gerstner received his Ph.D. degree in theoretical physics from the TU Munich, Germany, in 1993, after studies in Tübingen, Berkeley, and Munich. He is a Professor and Head of the Laboratory of Computational Neuroscience, EPFL, Switzerland.
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Wulfram Gerstner received his Ph.D. degree in theoretical physics from the TU Munich, Germany, in 1993, after studies in Tübingen, Berkeley, and Munich. He is a Professor and Head of the Laboratory of Computational Neuroscience, EPFL, Switzerland.
Laurent Badel is a Ph.D. student at the Laboratory of Computational Neuroscience (LCN) at the EPFL. He received his M.Sc. in physics from the EPFL in march 2003. His current research interests are mathematical and statistical approaches to the analysis of neuronal systems.
Magnus Richardson obtained his D-Phil in theoretical physics in 1997 at the University of Oxford. Since then he has worked on various subjects in the neurosciences and is currently interested in how neuronal properties lead to emergent states at the level of neural networks.