Spectral signatures of activity-dependent neural feedback in the corticothalamic system

The modulation of neural quantities by presynaptic and postsynaptic activities via local feedback processes is investigated by incorporating nonlinear phenomena such as relative refractory period, synaptic enhancement, synaptic depression, and habituation. This is done by introducing susceptibilities, which quantify the response in either firing threshold or synaptic strength to unit change in either presynaptic or postsynaptic activity. Effects on the power spectra are then analyzed for a realistic corticothalamic model to determine the spectral signatures of various nonlinear processes and to what extent these are distinct. Depending on the feedback processes, there can be enhancements or reductions in low-frequency and/or alpha power, splitting of the alpha resonance, and/or appearance of new resonances at high frequencies. These features in the power spectra allow processes to be fully distinguished where they are unique, or partly distinguished if they are common to only a subset of feedbacks, and can potentially be used to constrain the types, strengths, and dynamics of feedbacks present.

Figure 1

Schematic showing how the modulation of neural quantities of a postsynaptic neuron $a$ via local feedback processes is parametrized by each of the susceptibilities. The modulation of the firing threshold ${\theta }_a$ of neuron $a$ by the postsynaptic firing rate $Q_a$ is parametrized by the susceptibility ${\chi }_a$, and its modulation by presynaptic activity ${\varphi }_b$ is captured by ${\mu }_{ab}$. Modulation of the synaptic strength ${\nu }_{ab}$ by the postsynaptic firing rate $Q_a$ is parametrized by ${\eta }_{ab}$, and its modulation by presynaptic activity ${\varphi }_b$ is parametrized by ${\zeta }_{ab}$. Examples of underlying physiological effects are discussed in Secs. 2b and 2c.

Figure 2

Schematic of the corticothalamic model showing the connections between the system populations. The neural populations shown are cortical excitatory $e$ and inhibitory $i$, thalamic reticular $r$, and thalamic relay (or specific) nuclei $s$. Each parameter ${\nu }_{ab}$ quantifies the connection to population $a$ from population $b$. Excitatory connections are shown with solid arrows, and inhibitory connections are shown with dashed arrows.

Figure 3

The effect of ${\chi }_a$ on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for (a) $\eta =1000{\mathrm{s}}^{-1}$, (b) $\eta =100{\mathrm{s}}^{-1}$, and (c) $\eta =10{\mathrm{s}}^{-1}$. The black curves are for the baseline case ${\chi }_a=0$. Green (light gray) shows positive ${\chi }_a$ and red (dark gray) shows negative ${\chi }_a$. The dotted lines are for ${\chi }_a=\pm 0.3$ mV s and solid are for $\pm 0.8$ mV s.

Figure 4

Locus of $\det \mathbf{A}(\mathbf{k},\omega )$ in the complex plane for ${\chi }_a=-0.8$ mV s, $\eta =1000{\mathrm{s}}^{-1}$, $\left|\mathbf{k}\right|=0$ (solid line), and $\left|\mathbf{k}\right|\approx 28$ and $106{\mathrm{m}}^{-1}$ (dashed and dotted lines, respectively). Frequency $\omega$ increases counterclockwise as indicated by the arrows from $\omega \approx 0.6{\mathrm{s}}^{-1}$ (black square) to $\omega \approx 628{\mathrm{s}}^{-1}$ (white square). Strong resonances occur in the corresponding power spectrum when the locus is close the origin. Red crosses show the location of the alpha resonance for different spatial eigenmodes, blue crosses indicate the beta resonance, and, green crosses show the new resonance at approximately 90 Hz as illustrated in Fig. 3. In the inset, we observe that the loci diverge around the alpha resonance range, while they remain close around the other two resonances. The shift of the alpha resonance for individual spatial eigenmodes produces the shoulder and split alpha peak observed in the power spectrum calculated as per Eq. (40).

Figure 5

The effect of only ${\chi }_e$ on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for (a) $\eta =1000{\mathrm{s}}^{-1}$, (b) $\eta =100{\mathrm{s}}^{-1}$, and (c) $\eta =10{\mathrm{s}}^{-1}$. The black curves are for the baseline case ${\chi }_e=0$. Green (light gray) shows positive ${\chi }_e$ and red (dark gray) shows negative ${\chi }_e$. The dotted lines are for ${\chi }_e=\pm 0.08$ mV s and solid are for $\pm 0.3$ mV s.

Figure 6

Effect of ${\chi }_i$ only on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for $\eta =1000{\mathrm{s}}^{-1}$. The black curve is the baseline spectrum (${\chi }_i=0$). Green (light gray) shows positive ${\chi }_i$ and red (dark gray) shows negative ${\chi }_i$. The dotted lines are for ${\chi }_i=\pm 0.08$ mV s and solid are for $\pm 0.3$ mV s.

Figure 7

The effect of ${\mu }_{ab}$ (all individual ${\mu }_{ab}$ are equal) on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for $\eta =1000{\mathrm{s}}^{-1}$. The black curve is for the baseline case ${\mu }_{ab}=0$. Green shows positive ${\mu }_{ab}$ and red shows negative ${\mu }_{ab}$. The dotted lines are for ${\mu }_{ab}=\pm 8\mu \mathrm{V}$ s and solid are for $\pm 30\mu \mathrm{V}$ s.

Figure 8

The effect of ${\eta }_{ab}$ (all individual ${\eta }_{ab}$ are equal) on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for $\eta =1000{\mathrm{s}}^{-1}$. The black curve is for the baseline case ${\eta }_{ab}=0$. Green shows positive ${\eta }_{ab}$ and red shows negative ${\eta }_{ab}$. The dotted lines are for ${\eta }_{ab}=\pm 8\mu \mathrm{V}{\mathrm{s}}^2$ and solid are for $\pm 30\mu \mathrm{V}{\mathrm{s}}^2$.

Figure 9

The effect of ${\zeta }_{ab}$ (all individual ${\zeta }_{ab}$ are equal) on power spectra in the waking eyes-closed (EC) state using parameters from Table 1 for $\eta =1000{\mathrm{s}}^{-1}$. The black curve is for the baseline case ${\zeta }_{ab}=0$. Green shows positive ${\zeta }_{ab}$ and red shows negative ${\zeta }_{ab}$. The dotted lines are for ${\zeta }_{ab}=\pm 3\mu \mathrm{V}{\mathrm{s}}^2$ and solid are for $\pm 8\mu \mathrm{V}{\mathrm{s}}^2$.