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Axonal velocity distributions in neural field equations.

Bojak I, Liley DT - PLoS Comput. Biol. (2010)

Bottom Line: Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling.On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator.Our analytic results are also corroborated numerically using simulations on a large spatial grid.

View Article: PubMed Central - PubMed

Affiliation: Donders Institute for Brain, Cognition and Behaviour, Centre for Neuroscience, Radboud University Nijmegen, Nijmegen, The Netherlands. i.bojak@donders.ru.nl

ABSTRACT
By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

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Dispersive propagator: synaptic connectivity and marginal velocity distribution.(A) Synaptic connectivity  for different powers , which has been adjusted to match an exponential decay (thin curve). While the curves are continuous here, adjustment with Eq. (25) assumes a bin size , see text for details. (B) Marginal velocity distribution  for different powers . Note that concerning the dimensionless ratio  one obtains . The long-wavelength approximation  of Eq. (36) is shown for comparison as thin curve. See Eqns. (19) and (32) for (A) and (B), respectively.
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pcbi-1000653-g001: Dispersive propagator: synaptic connectivity and marginal velocity distribution.(A) Synaptic connectivity for different powers , which has been adjusted to match an exponential decay (thin curve). While the curves are continuous here, adjustment with Eq. (25) assumes a bin size , see text for details. (B) Marginal velocity distribution for different powers . Note that concerning the dimensionless ratio one obtains . The long-wavelength approximation of Eq. (36) is shown for comparison as thin curve. See Eqns. (19) and (32) for (A) and (B), respectively.

Mentions: We find that the normalization correction has an asymptotic value for large powers , whereas the decay correction grows as . The resulting synaptic connectivity is shown in Fig. 1A. For simplicity we have assumed here that , i.e., that . The dispersive curves are hence with the scaling factors derived above. While we show continuous curves here, the correction was performed for binned data. It is obvious from the reasonably close match that dispersive connectivity may well be mistaken for an exponential decay, given the large statistical and systematic errors typically involved in synaptic counts. Note that the divergence for small distances would not be visible in a binned count. Nevertheless, it is obvious that the case, and hence the long-wavelength approximation, does not match an exponential decay better than higher powers of . Furthermore, for in this example we find the optimal scaling and . In general for long-wavelength models one should actually choose and which are significantly larger than those measured in experiments. Note that our long-wavelength decay scale absorbed an expansion factor to keep Eq. (6) simple. Without this, scaling by would be required here.


Axonal velocity distributions in neural field equations.

Bojak I, Liley DT - PLoS Comput. Biol. (2010)

Dispersive propagator: synaptic connectivity and marginal velocity distribution.(A) Synaptic connectivity  for different powers , which has been adjusted to match an exponential decay (thin curve). While the curves are continuous here, adjustment with Eq. (25) assumes a bin size , see text for details. (B) Marginal velocity distribution  for different powers . Note that concerning the dimensionless ratio  one obtains . The long-wavelength approximation  of Eq. (36) is shown for comparison as thin curve. See Eqns. (19) and (32) for (A) and (B), respectively.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2813262&req=5

pcbi-1000653-g001: Dispersive propagator: synaptic connectivity and marginal velocity distribution.(A) Synaptic connectivity for different powers , which has been adjusted to match an exponential decay (thin curve). While the curves are continuous here, adjustment with Eq. (25) assumes a bin size , see text for details. (B) Marginal velocity distribution for different powers . Note that concerning the dimensionless ratio one obtains . The long-wavelength approximation of Eq. (36) is shown for comparison as thin curve. See Eqns. (19) and (32) for (A) and (B), respectively.
Mentions: We find that the normalization correction has an asymptotic value for large powers , whereas the decay correction grows as . The resulting synaptic connectivity is shown in Fig. 1A. For simplicity we have assumed here that , i.e., that . The dispersive curves are hence with the scaling factors derived above. While we show continuous curves here, the correction was performed for binned data. It is obvious from the reasonably close match that dispersive connectivity may well be mistaken for an exponential decay, given the large statistical and systematic errors typically involved in synaptic counts. Note that the divergence for small distances would not be visible in a binned count. Nevertheless, it is obvious that the case, and hence the long-wavelength approximation, does not match an exponential decay better than higher powers of . Furthermore, for in this example we find the optimal scaling and . In general for long-wavelength models one should actually choose and which are significantly larger than those measured in experiments. Note that our long-wavelength decay scale absorbed an expansion factor to keep Eq. (6) simple. Without this, scaling by would be required here.

Bottom Line: Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling.On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator.Our analytic results are also corroborated numerically using simulations on a large spatial grid.

View Article: PubMed Central - PubMed

Affiliation: Donders Institute for Brain, Cognition and Behaviour, Centre for Neuroscience, Radboud University Nijmegen, Nijmegen, The Netherlands. i.bojak@donders.ru.nl

ABSTRACT
By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Show MeSH
Related in: MedlinePlus