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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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Fits to binned counts of unmyelinated fibre diameters in rat subcortical white matter.The binned diameter data are averages over the unmyelinated data shown in Figs. 4–6 of Partadiredja et al. [36].  (magenta error bars) has been assumed to reflect mostly differential shrinkage, but fit dependence on this is mild. Fit results using the difference Eq. (57), and its dispersive counterpart Eq. (58), are collected in Tab. 4. For unmyelinated axons the optimal fit with ,  is shown. For comparison, the optimal  fit with the dispersive propagator is also displayed. It is viable, but has a three times larger . For the long-wavelength propagator a reasonable fit with Eq. (59) to all data cannot be obtained. Two curves are shown for illustration: one matching the median velocity of the difference ,  case, the other fitting only the first four data points.
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pcbi-1000653-g006: Fits to binned counts of unmyelinated fibre diameters in rat subcortical white matter.The binned diameter data are averages over the unmyelinated data shown in Figs. 4–6 of Partadiredja et al. [36]. (magenta error bars) has been assumed to reflect mostly differential shrinkage, but fit dependence on this is mild. Fit results using the difference Eq. (57), and its dispersive counterpart Eq. (58), are collected in Tab. 4. For unmyelinated axons the optimal fit with , is shown. For comparison, the optimal fit with the dispersive propagator is also displayed. It is viable, but has a three times larger . For the long-wavelength propagator a reasonable fit with Eq. (59) to all data cannot be obtained. Two curves are shown for illustration: one matching the median velocity of the difference , case, the other fitting only the first four data points.

Mentions: We now turn to animals, where more comprehensive data is available. Partadiredja et al. [36] have recently provided extensive data on axon diameter distributions in rat. As mentioned above, there appears to be a general trend in lower mammals that less of the small diameter fibres are myelinated. Rat data hence provides a convenient test for our difference propagator constructed to deal with such depletion, since we would expect more small diameter fibres and hence easier fits for other animals closer to humans, for example macaques. Furthermore, unlike the human data used above and other data sets from animals, Ref. [36] resolves diameters very finely and hence allows us to pinpoint the strengths and weaknesses of our ansatz. Partadiredja et al. [36] provide their fibre count data in terms of the total densities of axons per and corresponding percentage histograms dependent on fibre diameters, see their Tabs. 1 and 2 and Figs. 4, 5 and 6. They provide electromicroscopic results averaged over six adult male Wistar rats, but differentiated according to myelinated and unmyelinated axons of frontal, parietal, and occipital subcortical white matter from both left and right hemispheres.


Axonal velocity distributions in neural field equations.

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

Fits to binned counts of unmyelinated fibre diameters in rat subcortical white matter.The binned diameter data are averages over the unmyelinated data shown in Figs. 4–6 of Partadiredja et al. [36].  (magenta error bars) has been assumed to reflect mostly differential shrinkage, but fit dependence on this is mild. Fit results using the difference Eq. (57), and its dispersive counterpart Eq. (58), are collected in Tab. 4. For unmyelinated axons the optimal fit with ,  is shown. For comparison, the optimal  fit with the dispersive propagator is also displayed. It is viable, but has a three times larger . For the long-wavelength propagator a reasonable fit with Eq. (59) to all data cannot be obtained. Two curves are shown for illustration: one matching the median velocity of the difference ,  case, the other fitting only the first four data points.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC2813262&req=5

pcbi-1000653-g006: Fits to binned counts of unmyelinated fibre diameters in rat subcortical white matter.The binned diameter data are averages over the unmyelinated data shown in Figs. 4–6 of Partadiredja et al. [36]. (magenta error bars) has been assumed to reflect mostly differential shrinkage, but fit dependence on this is mild. Fit results using the difference Eq. (57), and its dispersive counterpart Eq. (58), are collected in Tab. 4. For unmyelinated axons the optimal fit with , is shown. For comparison, the optimal fit with the dispersive propagator is also displayed. It is viable, but has a three times larger . For the long-wavelength propagator a reasonable fit with Eq. (59) to all data cannot be obtained. Two curves are shown for illustration: one matching the median velocity of the difference , case, the other fitting only the first four data points.
Mentions: We now turn to animals, where more comprehensive data is available. Partadiredja et al. [36] have recently provided extensive data on axon diameter distributions in rat. As mentioned above, there appears to be a general trend in lower mammals that less of the small diameter fibres are myelinated. Rat data hence provides a convenient test for our difference propagator constructed to deal with such depletion, since we would expect more small diameter fibres and hence easier fits for other animals closer to humans, for example macaques. Furthermore, unlike the human data used above and other data sets from animals, Ref. [36] resolves diameters very finely and hence allows us to pinpoint the strengths and weaknesses of our ansatz. Partadiredja et al. [36] provide their fibre count data in terms of the total densities of axons per and corresponding percentage histograms dependent on fibre diameters, see their Tabs. 1 and 2 and Figs. 4, 5 and 6. They provide electromicroscopic results averaged over six adult male Wistar rats, but differentiated according to myelinated and unmyelinated axons of frontal, parietal, and occipital subcortical white matter from both left and right hemispheres.

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