Abstract
Phase response curves (PRCs) characterize response properties of oscillating neurons to pulsetile input and are useful for linking the dynamics of individual neurons to network dynamics. PRCs can be easily computed for model neurons. PRCs can be also measured for real neurons, but there are many issues that complicate this process. Most of these complications arise from the fact that neurons are noisy on several time-scales. There is considerable amount of variation (jitter) in the inter-spike intervals of “periodically” firing neurons. Furthermore, neuronal firing is not stationary on the time scales on which PRCs are usually measured. Other issues include determining the appropriate stimuli to use and how long to wait between stimuli.In this chapter, we consider many of the complicating factors that arise when generating PRCs for real neurons or “realistic” model neurons. We discuss issues concerning the stimulus waveforms used to generate the PRC and ways to deal with the effects of slow time-scale processes (e.g. spike frequency adaption). We also address issues that are present during PRC data acquisition and discuss fitting “noisy” PRC data to extract the underlying PRC and quantify the stochastic variation of the phase responses. Finally, we describe an alternative method to generate PRCs using small amplitude white noise stimuli.
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Notes
- 1.
The time constants of the synaptic currents will be faster than those for the PSP because the current waveform is filtered due to the RC properties neuronal membrane. To find the time constants of the synaptic currents, one can adjust these time constants until the current stimulus induces a PSP waveform that adequately matches an actual PSP.
- 2.
In general, the iPRC is equivalent to the gradient of phase with respect to all state variables evaluated at all points along the limit cycle (i.e. it is a vector measuring the sensitivity to perturbations in any variable). However, because neurons are typically only perturbed by currents, the iPRC for neurons is usually taken to be the voltage component of this gradient \(\left ( \frac{\partial \phi } {\partial V }\right )\) evaluated along the limit cycle.
- 3.
A delta-function is a pulse with infinite height and zero width with an area of one. Injecting a delta-function current into a cell corresponds to instantaneously injecting a fixed charge into the cell, which results in an instantaneous jump in the cell’s membrane potential by a fixed amount.
- 4.
The definition of a convolution is \(g {_\ast} f(\psi ) = \int \nolimits \nolimits g(\psi - t)f(t)\mathrm{d}t = \int \nolimits \nolimits g(-(t - \psi ))f(t)\mathrm{d}t\), so technically, \(PRC(\phi ) = Z {_\ast} l(-\phi T)\).
- 5.
Note that this could be done for polynomial fits too by using orthogonal polynomials (e.g., Legrendre or Chebychev polynomials).
- 6.
This formula for the AIC assumes that errors are independently distributed and described by a Gaussian distribution.
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Acknowledgments
TJL and MAS were supported by the National Science Foundation under grants DMS-09211039 (TJL), DMS-0518022 (TJL), TIN was supported by NSF CAREER Award CBET-0954797 (TIN).
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Netoff, T., Schwemmer, M.A., Lewis, T.J. (2012). Experimentally Estimating Phase Response Curves of Neurons: Theoretical and Practical Issues. In: Schultheiss, N., Prinz, A., Butera, R. (eds) Phase Response Curves in Neuroscience. Springer Series in Computational Neuroscience, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0739-3_5
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