Predictions and simulations of cortical dynamics during natural sleep using a continuum approach

M. T. Wilson, M. L. Steyn-Ross, D. A. Steyn-Ross, and J. W. Sleigh

Phys. Rev. E 72, 051910 – Published 7 November 2005

Abstract

In this paper we use a continuum model in two spatial dimensions to study the dynamics of the cortex during natural sleep, including explicitly the effects of two key neuromodulators. The model predicts that a number of states could be available to the cortex. We identify two of these with slow-wave sleep and rapid eye movement (REM) sleep, and focus on the transition between the two. Eigenvalue analysis of the linearized model, together with simulations on a two-dimensional grid, show that a number of oscillatory states exist; the occurrence of these is particularly dependent upon the duration in time of the inhibitory postsynaptic potential. These oscillatory states are similar to the cortical slow oscillation and certain types of seizure. Power spectra are evaluated for different parameter sets and compare favorably with experiment. Grid simulations show that transitions between cortical states (e.g., slow-wave to REM) can be seeded at any point in space by random fluctuations in subcortical input.

10 More

Received 20 January 2005

DOI:https://doi.org/10.1103/PhysRevE.72.051910

Authors & Affiliations

M. T. Wilson^1,*, M. L. Steyn-Ross¹, D. A. Steyn-Ross¹, and J. W. Sleigh²

¹Department of Physics and Electronic Engineering, Private Bag 3105, University of Waikato, Hamilton, New Zealand
²Department of Anaesthetics, Waikato Hospital, Hamilton, New Zealand

^*Electronic address: m.wilson@waikato.ac.nz;URL phys.waikato.ac.nz∕cortex

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 72, Iss. 5 — November 2005

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) A plot of $V_{e}$ for the stationary states in the sleep domain, using the standard parameter set of Table . Note how there are multiple stationary states for a region of the sleep domain at negative $Δ V_{e}^{rest}$ . The thick line on the diagram marks the turning points (where the gradient is infinite). Source: Modified from Ref. 17.Reuse & Permissions
Figure 2
An example of a grid used in the simulations. The rectangular cortex area $L_{x} \times L_{y}$ (solid box) is partitioned into $N$ gridsquares, each of area $Δ x Δ y$ . Each gridsquare contains a large number $(Δ x Δ y ∕ a_{mc})$ of macrocolumns; the bottom left square shows an example. A single gridpoint is placed at the center of each gridsquare, indicated by the crosses. The continuous function $\vec{y} (\vec{r}, t)$ is represented in a discrete form by its values at each gridpoint at a time $t$ ; the value of $\vec{y}$ at each gridpoint in effect represents the mean behavior of the macrocolumns over the gridsquare.Reuse & Permissions
Figure 3
(Color online) A plot of the sleep domain showing a region of multiple stationary states and a region of a single, unstable stationary state, for the parameters of Table . The instability of the point marked ⊕ is predicted in Fig. 4 and demonstrated in Fig. 6. The point marked ⊗ lies outside the lake of instability; its stability is demonstrated in Fig. 5.Reuse & Permissions
Figure 4
A plot of the real part of the largest eigenvalue pair against $q$ , taken in the unstable region of the domain (point ⊕ in Fig. 3). In this case, for small $q$ the real part of the dominant eigenvalue is positive, denoting an instability.Reuse & Permissions
Figure 5
A plot of (a) the excitatory and (b) the inhibitory firing rates for two locations in space (thick and thin lines) for the point ⊗ ( $Δ V_{e}^{rest} = 5.0 mV$ , $λ = 0.5$ ) outside the unstable lake of Fig. 3. In this case the system is stable to random noise input and never deviates far from the stationary state (shown by the dashed horizontal line). There is little spatial correlation but there is some correlation between $e$ and $i$ firing rates.Reuse & Permissions
Figure 6
A plot of the excitatory (solid line) and inhibitory (dashed line) firing rates against time, for the point ⊕ ( $Δ V_{e}^{rest} = 5.0 mV$ , $λ = 0.7$ ) in the unstable lake of Fig. 3. A small perturbation from the intial stationary state grows until the system reaches a limit cycle of frequency $\sim 4 Hz$ .Reuse & Permissions
Figure 7
The magnitude of the inhibitory postsynaptic potential impulse response against time. The three curves (solid, dashed, and dotted) correspond to values of $γ_{i}$ of 65, 42, and $15 s^{- 1}$ , respectively. All three have the same area $(= ρ_{i})$ but the largest $γ_{i}$ results in the most rapid transfer of charge across the synaptic junction.Reuse & Permissions
Figure 8
(Color online) A plot of the stability of the sleep domain for a decrease in $γ_{i}$ to $15 s^{- 1}$ . The unstable lake on the right has now grown into the region of multiple stationary states. Five possibilities exist: One stable state; one unstable state; three unstable states; one stable and two unstable states, and two stable and one unstable state (the midbranch is always unstable). The unmarked region of the figure corresponds to a single, stable state. See Fig. 9 for a simulation at the point ⊕ [three unstable states, coordinate ( $Δ V_{e}^{rest} = - 1.8 mV$ , $λ = 1.15$ )].Reuse & Permissions
Figure 9
A plot of (a) the excitatory and (b) the inhibitory firing rates at two locations on the cortex (black and gray lines) as a function of time, at a point in the sleep domain where there are multiple stationary states, but all are unstable. This is the point labelled ⊕ in Fig. 8, namely ( $Δ V_{e}^{rest} = - 1.8 mV$ , $λ = 1.15$ , $γ_{i} = 15 s^{- 1}$ ). The cortex exhibits a pulsing rhythm with the excitatory and inhibitory neuron firing rates being in phase. The cortex also rapidly synchronizes itself in space, with the firing rates at the two different points becoming more in phase as time increases.Reuse & Permissions
Figure 10
(a) A plot of the excitatory and inhibitory input fluxes ( $Φ_{e e}$ and $Φ_{i i}$ ) for two different random noise configurations. In both cases the system is started on the unstable upper branch (marked × on the left-hand panel). In one case the noise triggers the system to collapse onto the stable lower branch (thin line); in the other case the system enters a limit cycle that orbits the stationary states (thick line). The right panel (b) is an enlargement of the left, in the vicinity of the lower branch. This plot is for $γ_{i} = 15 s^{- 1}$ at the point ( $Δ V_{e}^{rest} = - 1.8 mV$ , $λ = 1.075$ ) in the sleep domain.Reuse & Permissions
Figure 11
The firing rates against time as the parameter $Δ V_{e}^{rest}$ is varied. The top graph (a) shows a plot of the stationary state solution for $V_{e}$ against $Δ V_{e}^{rest}$ for $λ = 1.26 mV$ . The upper branch is stable throughout, the lower and midbranches are unstable for the region shown on the graph. The bottom graph (b) shows a simulation in which $Δ V_{e}^{rest}$ is slowly raised. The time axis of this graph corresponds with the $Δ V_{e}^{rest}$ axis of the top graph. Initially (at $t = 0$ ), the system is started on the midbranch at the point ( $Δ V_{e}^{rest} = - 2.5 mV$ , $λ = 1.26$ , $γ_{i} = 15 s^{- 1}$ ) in the sleep domain and rapidly establishes a limit cycle. As time progresses, $Δ V_{e}^{rest}$ is increased at a constant rate so that the system moves into part of the sleep domain where there is a only single stable state. The system continues to oscillate. These oscillations are only quenched when $Δ V^{rest}$ is increased still further.Reuse & Permissions
Figure 12
A snapshot of a spiral wave generated by the cortical model (white is high $V_{e}$ , black is low $V_{e}$ ). In order to trigger the formation of spirals, the system was started on the stable region of the upper branch ( $Δ V_{e}^{rest} = 0.5 mV$ , $λ = 1.75$ , $γ_{i} = 33 s^{- 1}$ ) and $λ$ was rapidly lowered (over approximately $7 seconds$ ) through the unstable region and into the region of the domain where just one, stable state exists. The resulting spiral wave is extremely persistent and requires a considerable further reduction in $λ$ in order to destroy it.Reuse & Permissions
Figure 13
(Color online) This figure shows the power spectrum predicted by Ornstein-Uhlenbeck analysis either side of the lower- to upper-branch transition at $λ = 1.26$ with $γ_{i} = 65 s^{- 1}$ . Note that the vertical scale of the plots is different. The first plot (top left) shows the power spectrum on the lower branch at $Δ V_{e}^{rest} = - 4 mV$ ; we identify this with slow-wave sleep. The second and third plots show how the spectrum changes on the approach to transition. The final plot (bottom right) shows the power spectrum on the upper branch at $Δ V_{e}^{rest} = - 1 mV$ ; we identify this with REM sleep. The slow-wave plot shows a large power buildup at the lowest frequencies. Low spatial wave vectors indicate that the cortex has considerable synchronization over space. The REM sleep shows a much broader range of frequencies with lower powers than for slow-wave.Reuse & Permissions
Figure 14
EEGs recorded from a single subject in [(a), left] slow-wave and [(b), right] REM sleep (solid lines). [Data courtesy of Fisher and Paykel Healthcare—the patient having consented for data to be used for research purposes.] The data have been clipped at $1 Hz$ to remove artifacts from the recording process and have been smoothed. The dotted lines here represent the simulated $S (q = 0, ω)$ of Fig. 13 scaled by an appropriate factor. The slow-wave has been represented by a point on the lower branch close to transition, namely ( $λ = 1.25$ , $Δ V_{e}^{rest} = - 1.8 mV$ , $γ_{i} = 65 s^{- 1}$ ). REM has been represented by a point with virtually identical parameters but on the upper branch, namely ( $λ = 1.26$ , $Δ V_{e}^{rest} = - 1.8 mV$ , $γ_{i} = 65 s^{- 1}$ ). Although the slow-wave case shows a good fit to the data, the mapping between $S (q, ω)$ and EEG is complicated and not discussed further in this paper.Reuse & Permissions
Figure 15
(Color online) The predicted and simulated power spectra for the point ⊗ of Fig. 3, ( $Δ V_{e}^{rest} = 5.0 mV$ , $λ = 0.5$ ). The first row shows the predicted (a) and simulated spectra for $γ_{i} = 65 s^{- 1}$ . The second shows predicted (c) and simulated (d) spectra for $γ_{i} = 53 s^{- 1}$ , and the third predicted (e) and simulated (f) spectra for $γ_{i} = 46 s^{- 1}$ . In the case of (e) and (f) the unstable lake has grown so that our simulation lies almost on the edge of the bifurcation. Note the vertical scales of the plots are not the same. In all cases the linearized prediction and simulation are in very close agreement.Reuse & Permissions
Figure 16
(Color online) The excitatory soma potential as a function of time, for the point ( $Δ V_{e}^{rest} = - 1.8 mV$ , $λ = 1.25$ , $γ_{i} = 65 s^{- 1}$ ). The left-hand diagram shows three locations as a plot of $V_{e}$ against time (specifically the point of initial transition and points $12.5 cm$ and $25 cm$ from this); the right-hand shows a gray scale plot of a slice through space against time (black is low potential, white is high). Here the cortex starts on the lower branch of a multiple-state region, but very close to the transition point. Noise fluctuations eventually trigger one location on the cortex (close to the zero distance on the space axis) to move onto the upper branch. Once this transition has occurred, the rest of the cortex is dragged onto the upper branch. The complete transition takes about $0.5 s$ . Note that there is initially an overshoot in $V_{e}$ as the system moves into the upper state.Reuse & Permissions
Figure 17
(Color online) A plot of excitatory soma potential against time for locations in space for the point ( $Δ V_{e}^{rest} = - 1.0 mV$ , $λ = 1.123$ , $γ_{i} = 41 s^{- 1}$ ). The left-hand diagram shows two locations as a plot of $V_{e}$ against time; the right-hand shows a gray scale plot of a slice through space against time (black is low potential, white is high). The cortex is in a region of multiple stationary states; the lower state is unstable but the upper is stable. Initially the cortex is placed on the lower branch and it quickly enters a limit cycle, since this branch is unstable. However, at one location the cortex becomes trapped in the upper state. The rest of the cortex continues to pulse. The trapped area of space slowly grows with time; eventually the whole cortex has been drawn into the stable upper state.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics