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Spiking neural network simulation: numerical integration with the Parker-Sochacki method.

Stewart RD, Bair W - J Comput Neurosci (2009)

Bottom Line: The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models.We apply the Parker-Sochacki method to the Izhikevich 'simple' model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods.Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, OX1 3PT, UK. Robert.Stewart@pharm.ox.ac.uk

ABSTRACT
Mathematical neuronal models are normally expressed using differential equations. The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models. Using this method, the solution order can be adapted according to the local conditions at each time step, enabling adaptive error control without changing the integration timestep. The method has been limited to polynomial equations, but we present division and power operations that expand its scope. We apply the Parker-Sochacki method to the Izhikevich 'simple' model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods. Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

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Izhikevich model recurrent network results: recordings from repeated simulations using the same network model and initial random inputs but varying the integration method and error tolerance/time step size. (a) Membrane potential traces from single neuron. (Top to bottom) Conditions 1–3. In each plot, the colour represents the integration method, with an additional reference trace drawn in black. The reference trace was drawn last. Thus, the appearance of coloured lines indicates divergence of test solutions from the reference. (b) Population raster plot of the first twenty cells in the same network model as in (a); layout and colours as in (a). The single cell traced in (a) appears here as neuron 5. As in (a), these plots are overlain by a reference solution in black. Thus, the appearance of coloured dots indicates divergence
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Fig3: Izhikevich model recurrent network results: recordings from repeated simulations using the same network model and initial random inputs but varying the integration method and error tolerance/time step size. (a) Membrane potential traces from single neuron. (Top to bottom) Conditions 1–3. In each plot, the colour represents the integration method, with an additional reference trace drawn in black. The reference trace was drawn last. Thus, the appearance of coloured lines indicates divergence of test solutions from the reference. (b) Population raster plot of the first twenty cells in the same network model as in (a); layout and colours as in (a). The single cell traced in (a) appears here as neuron 5. As in (a), these plots are overlain by a reference solution in black. Thus, the appearance of coloured dots indicates divergence

Mentions: In this section, we characterise the outputs from a single experiment, using the reference solution to assess accuracy. Figure 3(a) shows membrane potential traces from a single neuron, with results from conditions 1–3 arranged in separate panels, top to bottom and integration methods represented using different colours. For comparison, the trace from the reference solution for this experiment is plotted as a black line in each panel. The reference trace was drawn last so that it would obscure the coloured traces when they were in agreement. Thus, working left to right in a single panel, the appearance of a coloured line is a visual indicator of divergence between the reference solution and the test solution from the method represented by that colour. Fig. 3


Spiking neural network simulation: numerical integration with the Parker-Sochacki method.

Stewart RD, Bair W - J Comput Neurosci (2009)

Izhikevich model recurrent network results: recordings from repeated simulations using the same network model and initial random inputs but varying the integration method and error tolerance/time step size. (a) Membrane potential traces from single neuron. (Top to bottom) Conditions 1–3. In each plot, the colour represents the integration method, with an additional reference trace drawn in black. The reference trace was drawn last. Thus, the appearance of coloured lines indicates divergence of test solutions from the reference. (b) Population raster plot of the first twenty cells in the same network model as in (a); layout and colours as in (a). The single cell traced in (a) appears here as neuron 5. As in (a), these plots are overlain by a reference solution in black. Thus, the appearance of coloured dots indicates divergence
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2717378&req=5

Fig3: Izhikevich model recurrent network results: recordings from repeated simulations using the same network model and initial random inputs but varying the integration method and error tolerance/time step size. (a) Membrane potential traces from single neuron. (Top to bottom) Conditions 1–3. In each plot, the colour represents the integration method, with an additional reference trace drawn in black. The reference trace was drawn last. Thus, the appearance of coloured lines indicates divergence of test solutions from the reference. (b) Population raster plot of the first twenty cells in the same network model as in (a); layout and colours as in (a). The single cell traced in (a) appears here as neuron 5. As in (a), these plots are overlain by a reference solution in black. Thus, the appearance of coloured dots indicates divergence
Mentions: In this section, we characterise the outputs from a single experiment, using the reference solution to assess accuracy. Figure 3(a) shows membrane potential traces from a single neuron, with results from conditions 1–3 arranged in separate panels, top to bottom and integration methods represented using different colours. For comparison, the trace from the reference solution for this experiment is plotted as a black line in each panel. The reference trace was drawn last so that it would obscure the coloured traces when they were in agreement. Thus, working left to right in a single panel, the appearance of a coloured line is a visual indicator of divergence between the reference solution and the test solution from the method represented by that colour. Fig. 3

Bottom Line: The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models.We apply the Parker-Sochacki method to the Izhikevich 'simple' model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods.Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

View Article: PubMed Central - PubMed

Affiliation: Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, OX1 3PT, UK. Robert.Stewart@pharm.ox.ac.uk

ABSTRACT
Mathematical neuronal models are normally expressed using differential equations. The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models. Using this method, the solution order can be adapted according to the local conditions at each time step, enabling adaptive error control without changing the integration timestep. The method has been limited to polynomial equations, but we present division and power operations that expand its scope. We apply the Parker-Sochacki method to the Izhikevich 'simple' model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods. Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

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