Hallucinogen persisting perception disorder in neuronal networks with adaptation

Abstract

We study the spatiotemporal dynamics of neuronal networks with spike frequency adaptation. In particular, we compare the effects of adaptation being either a linear or nonlinear function of neural activity. We find that altering parameters controlling the strength of synaptic connections in the network can lead to spatially structured activity suggestive of symptoms of hallucinogen persisting perception disorder (HPPD). First, we study how both networks track spatially homogeneous flickering stimuli, and find input is encoded as continuous at lower flicker frequencies when the network’s synapses exhibit more net excitation. Mainly, we study instabilities of stimulus-driven traveling pulse solutions, representative of visual trailing phenomena common to HPPD patients. Visual trails are reported as discrete afterimages in the wake of a moving input. Thus, we analyze several solutions arising in response to moving inputs in both networks: an ON state, stimulus-locked pulses, and traveling breathers. We find traveling breathers can arise in both networks when an input moves beyond a critical speed. These possible neural substrates of visual trails occur at slower speeds when the modulation of synaptic connectivity is increased.

Appendices

Appendix A: ON state in system with nonlinear adaptation

In this appendix, we derive Eq. (4.3), giving the critical speed c _* at which the ON state ceases to exist in the full network with nonlinear adaptation (2.2). Thus, we study the equations,

$$\begin{array}{rll} u_t &=& - u + \int_{- \pi}^{\pi} (w_0 + w_2 \cos (x-x')) \nonumber\\ && \times\,H (u(x',t)- v(x',t) - \kappa) {\rm d} x' + I_0 \cos^2 \left( \frac{x- ct}{2} \right), \nonumber \\ v_t &=& (-v + \beta H(u-v-\kappa))/ \alpha. \end{array}$$

(A.1)

Changing coordinates to the rotating frame ξ = x − ct, so u = U(ξ) and v = V(ξ), and assuming the ON state exits (U(ξ) > V(ξ) + κ), we have

$$\begin{array}{rll} -cU'( \xi ) &=& -U ( \xi ) + \int_{- \pi }^{\pi} w_0 + w_2 \cos ( \xi - \xi') {\rm d} \xi' \\ &&+\, I_0 \cos^2 \left( \frac{\xi}{2} \right), \\ -cV' ( \xi ) &=& (- V( \xi ) + \beta ) / \alpha. \end{array}$$

Solving this pair of first order differential equations by along with periodic boundary conditions yields

$$ U( \xi ) = 2 \pi w_0 + \frac{I_0}{2} + \frac{I_0 ( \cos \xi - c \sin \xi )}{2(1+ c^2)} $$

(A.2)

and V(ξ) = β. For the ON state to exist, synaptic input must be superthreshold, U(ξ) > κ + β for all ξ ∈ ( − π, π). Simply requiring this of the minimum, U(ξ _m ) > κ + β, yields the following inequality

$$ 2 \pi w_0 + \frac{I_0( \sqrt{1+ c^2} - 1)}{2 \sqrt{1+c^2}} > \beta + \kappa. $$

(A.3)

We then identify the critical stimulus speed value c _* at which the ON state ceases to exist as that for which the inequality (A.3) becomes a strict equality. Thus, we can solve the equality explicitly for Eq. (4.3):

$$ c_* = \sqrt{ \frac{I_0^2}{(4 \pi w_0 + I_0 - 2 ( \beta + \kappa ) )^2} - 1}, $$

so when c > c _*, the ON state exists.

Appendix B: Existence of pulses in network with nonlinear adaptation

Here, we present our construction of stimulus-locked traveling pulse solutions in the network with nonlinear adaptation (2.2) and Heaviside firing rate function (2.5). Changing to traveling wave coordinates ξ = x − ct, a pulse of width Δ ∈ (0, 2 π) has superthreshold region ξ ∈ (π − Δ, π) , where we can arbitrarily select the leading edge’s location due the translation invariance of the problem. We also specify a spatial shift Δ _I of the input $I(x,t) = I_0 \cos^2 ( ( \xi + \Delta_I)/2)$. This yields

$$\begin{array}{rll} -c U'( \xi ) &=& - U( \xi ) + \int_{\pi - \Delta}^{\pi} w( \xi - \xi') {\rm d} \xi' \\ &&+\, I_0 \cos^2 \left( \frac{\xi + \Delta_I}{2} \right), \\ -c V'( \xi ) &=& (- V( \xi ) + \beta \Theta (\xi ) )/ \alpha, \end{array}$$

where

$$ \Theta ( \xi ) = \left\{ \begin{array}{ll} 1, & \xi \in ( \pi - \Delta , \pi ), \\ 0, & \xi \in ( - \pi , \pi - \Delta ). \end{array} \right. $$

Solving this set of equations with periodic boundary conditions gives

$$\begin{array}{rll} U( \xi ) &=& w_0 \Delta + \frac{I_0 [ \cos ( \xi + \Delta_I ) - c \sin ( \xi + \Delta_I ) ]}{2 (1 + c^2)} \nonumber\\ &&+\, \frac{I_0}{2} + \frac{w_2 ( \sin \xi - \sin ( \xi + \Delta))}{1+ c^2} \nonumber\\ &&+\, \frac{w_2 c ( \cos \xi - \cos ( \xi + \Delta))}{1+ c^2}, \end{array}$$

(B.1)

$$\begin{array}{rll} V(\xi ) &=& \beta \left( 1 - \frac{\displaystyle {\rm e}^{\xi/ \alpha c} - {\rm e}^{(\xi + \Delta - 2 \pi)/ \alpha c}}{\displaystyle 2 \sinh ( \pi/ \alpha c )} \right) : \xi \in ( \pi - \Delta, \pi ), \\ V( \xi ) &=& \beta \left( \frac{\displaystyle {\rm e}^{(\xi + \Delta)/ \alpha c} - {\rm e}^{\xi / \alpha c}}{\displaystyle 2 \sinh (\pi/ \alpha c )} \right) : \xi \in (- \pi , \pi - \Delta ). \end{array}$$

(B.2)

Therefore, to specify the pulse width Δ and associated shift to the input Δ _I , we impose self consistency by requiring that the input current U(ξ) − V(ξ) specified by Eqs. (B.1) and (B.2) cross firing threshold κ at the leading and trailing edge of the pulse such that U(π) − V(π) = U(π − Δ) − V(π − Δ) = κ. This generates the following nonlinear system of equations

$$\begin{array}{lll} && w_0 \Delta + \frac{I_0}{2} + \frac{I_0 [c \sin \Delta_I - \cos \Delta_I]}{2(1+ c^2)} \\ &&\;+ \frac{w_2 [ \sin \Delta + c ( \cos \Delta - 1)]}{1+ c^2} = \beta \frac{{\rm e}^{\Delta/ \alpha c} - 1}{{\rm e}^{2 \pi/ \alpha c}-1} + \kappa, \\ && w_0 \Delta + \frac{I_0}{2} + \frac{I_0 [c \sin ( \Delta_I - \Delta ) - \cos ( \Delta_I - \Delta )]}{2(1+ c^2)} \\ &&\;+ \frac{w_2 [ \sin \Delta + c ( 1- \cos \Delta )]}{1+ c^2} = \beta \frac{1- {\rm e}^{-\Delta / \alpha c}}{1 - {\rm e}^{-2 \pi / \alpha c}} + \kappa. \end{array}$$

Appendix C: Stability of pulses in network with nonlinear adaptation

In this appendix, we derive the linear stability approximation for traveling pulses in the network with nonlinear adaptation. As in the case of the network with linear adaptation, we start by considering the evolution of small, smooth perturbations to the traveling pulse solution (U(ξ), V(ξ)), such that $u( \xi , t) = U( \xi ) + \bar{\psi} ( \xi , t)$ and $v( \xi , t) = V( \xi ) + \bar{\phi} ( \xi , t)$. Upon plugging these expression into the system (2.2) and expanding to first order in $( \bar{\psi} , \bar{\phi} )$, we arrive at the system of linear equations

$$\begin{array}{rll} \bar{\psi}_t - c \bar{\psi}_{\xi} + \bar{\psi} &=& \int_{-\pi}^{\pi} w( \xi - \xi') H'( U( \xi' ) - V( \xi ') - \kappa ) \notag\\ &&\times\, ( \bar{\psi} ( \xi',t) - \bar{\phi} ( \xi',t)) {\rm d} \xi', \\ \bar{\phi}_t -c \bar{\phi}_{\xi} + \frac{\bar{\phi}}{\alpha} &=& \frac{\beta}{\alpha} H' ( U - V - \kappa ) ( \bar{\psi} - \bar{\phi} ). \end{array}$$

To characterize some of the spectrum of this operator, we can look, in particular, for solutions of the form $( \bar{\psi}, \bar{\phi}) = {\rm e}^{\lambda t} ( \psi ( \xi ) , \phi ( \xi ))$ and using the identity

$$\begin{array}{rll} \frac{{\rm d} H(U - V - \kappa )}{{\rm d} U} &=& \frac{\delta ( \xi - \pi + \Delta )}{|U'(\pi- \Delta) - V'( \pi - \Delta ) |} \\ &&+ \frac{\delta ( \xi - \pi )}{|U'( \pi ) - V'( \pi )|}, \end{array}$$

(C.1)

where

$$\begin{array}{rll} U' ( \pi ) &=& \frac{w_2 [ \cos \Delta - 1 - c \sin \Delta ]}{1+c^2} \\ &&+ \frac{I_0 [ \sin \Delta_I + c \cos \Delta_I ]}{2 (1 + c^2)}, \\ U'( \pi - \Delta ) &=& \frac{w_2 [ 1- \cos \Delta - c \sin \Delta ]}{1 + c^2} \\ &&+ \frac{I_0 [ \sin ( \Delta_I - \Delta ) + c \cos ( \Delta_I - \Delta )]}{2 ( 1+ c^2)}. \end{array}$$

Due to the jump discontinuity in V′( ξ) at the threshold crossing points, either V′( π) = V ₊′( π) or V′(π) = V ₋ ′(π), and similarly either V′(π − Δ) = V ₊′ (π − Δ) or V′(π − Δ) = V ₋ ′(π − Δ) where

$$ V_+' ( \pi ) = - \frac{\beta (1 - {\rm e}^{(\Delta - 2 \pi )/ \alpha c})}{\alpha c ( 1- {\rm e}^{-2 \pi / \alpha c})}, $$

(C.2)

$$ V_-' ( \pi ) = \frac{\beta ( {\rm e}^{\Delta / \alpha c}-1)}{\alpha c( {\rm e}^{2 \pi / \alpha c} - 1)}, $$

(C.3)

$$ V_+' ( \pi - \Delta ) = - \frac{\beta ( {\rm e}^{- \Delta / \alpha c} - {\rm e}^{-2 \pi / \alpha c})}{\alpha c ( 1- {\rm e}^{-2 \pi / \alpha c})}, $$

(C.4)

$$ V_-'( \pi - \Delta ) = \frac{\beta ( 1 - {\rm e}^{- \Delta / \alpha c})}{\alpha c ( 1- {\rm e}^{-2 \pi / \alpha c})}. $$

(C.5)

Note that, following recent studies of the stability of spatially structured solutions in piecewise smooth neural fields (Kilpatrick and Bressloff 2010a, c; Bressloff and Kilpatrick 2011), the particular choice to make for each V′(π) and V′(π − Δ) will be assigned based the sign of the difference of perturbations (ψ(ξ) − ϕ(ξ)) in both places the expression appears in the identity (C.1). This is due to the piecewise smooth nature of V′ at the boundary of the pulse. Upon assuming a fixed sign of these perturbations, we can only calculate real eigenvalues associated with the resulting piecewise defined system. This is due to the fact that solutions ψ(ξ) − ϕ(ξ) with associated complex eigenvalues will oscillate above and below zero, violating our assumption of a fixed sign. Thus, we obtain the following system

$$\begin{array}{rll} -c \psi' + ( \lambda + 1) \psi &=& \chi_{\pi}w( \xi - \pi ) (\psi ( \pi ) - \phi ( \pi )) \\ &&+ \chi_{\pi - \Delta} w( \xi - \pi + \Delta ) \\ &&\times\,(\psi ( \pi - \Delta ) - \phi ( \pi - \Delta)) \nonumber \\ -c \phi' + ( \lambda + \alpha^{-1}) \phi &=& \frac{\beta}{\alpha} \big[ \chi_{\pi} \delta ( \xi - \pi ) (\psi ( \pi ) - \phi ( \pi )) \\ &&\quad+ \chi_{\pi - \Delta} \delta ( \xi - \pi + \Delta )\\ &&\quad\times\, (\psi ( \pi - \Delta) - \phi ( \pi - \Delta ))\big], \end{array}$$

(C.6)

where

$$ \chi_{\xi}^{-1} = \left\{ \begin{array}{ll} |U'( \xi ) - V_+'( \xi ) |, & {\rm if} \ \ \psi ( \xi ) < \phi ( \xi ), \\ |U'( \xi ) - V_-'( \xi ) |, & {\rm if} \ \ \psi ( \xi ) > \phi ( \xi ). \end{array} \right. $$

(C.7)

Therefore, the stability of the stimulus-locked traveling pulse given by Eqs. (B.1) and (B.2) can be calculated using spectrum of the eigenvalue problem (C.6). First, note that the values bounding the essential spectrum λ = − 1 and λ = − α ^− 1 have infinite multiplicity and will not contribute to any instabilities. Upon omitting these value from our analysis, we can proceed by solving for the eigenfunctions. It is straightforward to do so by directly integrating both equations, treating the ψ(π) , ψ(π − Δ), ϕ(π), ϕ(π − Δ) terms as constants, to find

$$\begin{array}{rll} \psi ( \xi ) &=& \chi_{\pi} \left[ {\mathcal P}_0 + \frac{{\mathcal P}_1 \sin \xi - {\mathcal P}_2 \cos \xi }{{\mathcal D}_p} \right] ( \psi ( \pi ) - \phi ( \pi ) ) \\ && + \chi_{\pi - \Delta} \left[ {\mathcal P}_0 + \frac{{\mathcal P}_1 \sin ( \xi + \Delta ) - {\mathcal P}_2 \cos ( \xi + \Delta )}{{\mathcal D}_p} \right] \\ &&\times\,( \psi ( \pi - \Delta ) - \phi ( \pi - \Delta )), \end{array}$$

(C.8)

$$\begin{array}{rll} \phi ( \xi ) &=& \frac{\beta}{\alpha c} \bigg\{ \chi_{\pi} \left[ \frac{1}{{\mathcal P}_3} - H( \xi - \pi ) \right] ( \psi ( \pi) - \phi ( \pi ) ) \\ && \,\,\quad + \chi_{\pi - \Delta} \left[ \frac{1}{{\mathcal P}_3} - H( \xi - \pi + \Delta ) \right] {\rm e}^{\frac{\alpha \lambda + 1}{\alpha c} \Delta} \\ &&\,\,\,\quad\times\,( \psi ( \pi - \Delta ) - \phi ( \pi - \Delta )) \bigg\} {\rm e}^{\frac{\alpha \lambda + 1}{\alpha c} ( \xi - \pi )}, \end{array}$$

(C.9)

where

$$\begin{array}{rll}{\mathcal P}_0 &=& \frac{w_0}{\lambda + 1}, \\ {\mathcal P}_1 &=& w_2 c, \\ {\mathcal P}_2 &=& w_2 ( \lambda + 1), \\ {\mathcal P}_3 &=& 1 - {\rm e}^{- \frac{2 ( \alpha \lambda + 1) \pi}{\alpha c}}, \\ {\mathcal D}_p &=& ( \lambda + 1)^2 + c^2 . \end{array}$$

Requiring self-consistency of the the solutions (C.8) and (C.9), we generate the following 2 × 2 system of equations ${\boldsymbol \psi} - {\boldsymbol \phi} = {\mathcal A}_p ({\boldsymbol \psi} - {\boldsymbol \phi})$ where

$$\begin{array}{rll}{\boldsymbol \psi} &=& \left( \begin{array}{c} \psi ( \pi ) \\ \psi ( \pi - \Delta ) \end{array} \right) , \ \ \ \ \ {\boldsymbol \phi} = \left( \begin{array}{c} \phi ( \pi ) \\ \phi ( \pi - \Delta ) \end{array} \right), \\ {\mathcal A}_p &=& \left( \begin{array}{cc} {\mathcal A}_{\pi \pi} & {\mathcal A}_{\pi \Delta} \\ {\mathcal A}_{\Delta \pi} & {\mathcal A}_{\Delta \Delta} \end{array} \right), \end{array}$$

(C.10)

with

$$\begin{array}{rll}{\mathcal A}_{\pi \pi } &=& \chi_{\pi} \left[ \frac{{\mathcal D}_p {\mathcal P}_0 + {\mathcal P}_2}{{\mathcal D}_p} - \frac{\beta}{\alpha c} \left( \frac{1}{{\mathcal P}_3} - {\mathcal H}_{\pi} \right) \right], \\ {\mathcal A}_{\pi \Delta} &=& \chi_{\pi - \Delta} \left[ \frac{{\mathcal D}_p {\mathcal P}_0 - {\mathcal P}_1 \sin \Delta + {\mathcal P}_2 \cos \Delta}{{\mathcal D}_p} \right.\\ &&\quad\qquad-\left. \frac{\beta}{\alpha c} \left[ \frac{1}{{\mathcal P}_3} - 1 \right] {\rm e}^{\frac{\alpha \lambda + 1}{\alpha c} \Delta} \right], \\ {\mathcal A}_{\Delta \pi} &=& \chi_{\pi} \left[ \frac{{\mathcal D}_p {\mathcal P}_0 + {\mathcal P}_1 \sin \Delta + {\mathcal P}_2 \cos \Delta}{{\mathcal D}_p} \right.\\ &&\qquad-\left. \frac{\beta}{\alpha c} \left( \frac{1}{{\mathcal P}_3} \right) {\rm e}^{- \frac{\alpha \lambda +1}{\alpha c} \Delta} \right], \\ {\mathcal A}_{\Delta \Delta} &=& \chi_{\pi -\Delta} \left[ \frac{{\mathcal D}_p {\mathcal P}_0 + {\mathcal P}_2}{{\mathcal D}_p} - \frac{\beta}{\alpha c} \left( \frac{1}{{\mathcal P}_3} - {\mathcal H}_{\pi -\Delta} \right) \right], \end{array}$$

and

$$\begin{array}{rll}{\mathcal H}_{\pi} &=& \left\{ \begin{array}{ll} 1, & {\rm if} \ \ \psi ( \pi ) > \phi ( \pi ), \\ 0, & {\rm if} \ \ \psi ( \pi ) < \phi ( \pi ), \end{array} \right. \\ {\mathcal H}_{\pi - \Delta} &=& \left\{ \begin{array}{ll} 1, & {\rm if} \ \ \psi ( \pi - \Delta ) < \phi ( \pi - \Delta ), \\ 0, & {\rm if} \ \ \psi ( \pi - \Delta ) > \phi ( \pi - \Delta ). \end{array} \right. \end{array}$$

Now in order to examine the stability, we look for nontrivial solutions of ${\boldsymbol \psi} - {\boldsymbol \phi} = {\mathcal A}_p ({\boldsymbol \psi} - {\boldsymbol \phi})$ such that ${\mathcal E}( \lambda ) = 0$, where ${\mathcal E}( \lambda ) = \det ( {\mathcal A}_p - I)$, is the Evans function of the traveling pulse solution (B.1) and (B.2). The traveling pulse will surely not be linearly stable in the case that λ > 0 for all real λ such that ${\mathcal E}( \lambda ) = 0$. We cannot ensure linear stability, since we have omitted eigensolutions with complex eigenvalues. However, we find that, in many instances, our analysis accurately predicts the nonlinear stability of the associated stimulus-locked traveling pulse. The Evans function is then piecewise defined for four possible classes of eigenfunction: (i) ψ(π) > ϕ(π) and ψ(π − Δ) > ϕ(π − Δ); (ii) ψ(π) > ϕ(π) and ψ(π − Δ) < ϕ(π − Δ); (iii) ψ(π) < ϕ(π) and ψ(π − Δ) > ϕ(π − Δ); and (iv) ψ(π) < ϕ(π) and ψ(π − Δ) < ϕ(π − Δ).