Delay- and Coupling-Induced Firing Patterns in Oscillatory Neural Loops

For a feedforward loop of oscillatory Hodgkin-Huxley neurons interacting via excitatory chemical synapses, we show that a great variety of spatiotemporal periodic firing patterns can be encoded by properly chosen communication delays and synaptic weights, which contributes to the concept of temporal coding by spikes. These patterns can be obtained by a modulation of the multiple coexisting stable in-phase synchronized states or traveling waves propagating along or against the direction of coupling. We derive explicit conditions for the network parameters allowing us to achieve a desired pattern. Interestingly, whereas the delays directly affect the time differences between spikes of interacting neurons, the synaptic weights control the phase differences. Our results show that already such a simple neural circuit may unfold an impressive spike coding capability.

Figure 1

(a) Ring of unidirectionally coupled neurons. (b)–(d) Typical periodic spatiotemporal patterns appearing in homogeneously coupled (identical synaptic weights and communication delays) HH spiking neurons (1): (b) in-phase synchronization, (c),(d)  traveling waves with one firing front propagating along ($k=1$) and against ($k=-1)$ the direction of coupling, respectively, see (a). Membrane potential $V_j$ is encoded in color versus time and oscillator number. (e) Coexistence of multiple stable traveling waves, where the spiking frequency of synchronized neurons is depicted versus the number of firing fronts $k$ for different identical delay $\tau$ in coupling and for initial vector functions $\Gamma (t)$, $t\in [-\tau ,0]$ with components ${\Gamma }_j(t)=\gamma (t+jTk/N)$, where $\gamma (t)$ approximates the waveform of the limit cycle of a neuron in the network, and $T$ is its period. Parameters $C=1\mu \mathrm{F}/{\mathrm{cm}}^2$, $V_{Na}=50\mathrm{mV}$, $V_K=-77\mathrm{mV}$, $V_l=-54.4\mathrm{mV}$, $g_{Na}=120\mathrm{mS}/{\mathrm{cm}}^2$, $g_K=36\mathrm{mS}/{\mathrm{cm}}^2$, and $g_l=0.3\mathrm{mS}/{\mathrm{cm}}^2$. The constant currents $I_j$ are Gaussian distributed with mean $10\mu \mathrm{A}/{\mathrm{cm}}^2$ and standard deviation 0.01. Delay ${\tau }_j=0\mathrm{ms}$ in (b)–(d), number of neurons $N=100$, and coupling $K_j=5\mathrm{mS}/{\mathrm{cm}}^2$.

Figure 2

Examples of delay-induced spatiotemporal patterns in the HH ensemble (1). (a),(c) Raster plots of the neural firing induced by delays ${\tau }_j$ depicted in (b) and (d), respectively. Green circles show the corresponding reference pattern of (a) in-phase synchronization and (c) traveling wave with $k=-1$ firing fronts for identical delays ${\tau }_j=5\mathrm{ms}$. Blue empty diamonds indicate spike onsets from numerical simulations, and red filled diamonds depict theoretically predicted patterns. To observe the above complex patterns the reference patterns were numerically continued by slowly approaching the predicted delays. Other parameters as in Fig. 1.

Figure 3

Examples of the coupling-induced spatiotemporal patterns of the LC (2) and HH (1) system. (a) $S$-shaped pattern in LC system induced by the coupling strengths $K_j$ from (b). (c) The corresponding amplitudes ${\rho }_j$ of LC oscillators. (d) Similar $S$-shaped pattern in the HH ensemble with the same coupling strengths. The membrane potential $V_j$ is encoded in color. (e) Stable “PRL” pattern induced in the HH ensemble by the inhomogeneous synaptic weights $K_j$ from (f). The weights are obtained from Eq. (3) when ignoring the amplitudes. Green circles in (a) and (e) indicate the in-phase synchronized reference pattern for homogeneous couplings $K_j=5$. The theoretically predicted patterns [black curve in (d) and red filled diamonds in plots  (a), (c), and (e)] fit well to the numerical simulations [(d) and blue empty diamonds in (a), (c), and (e)]. In (d) and (e) the theoretically predicted patterns are scaled by a factor of 13 to fit to the temporal scale of the firing pattern obtained in the HH ensemble. Parameters: $\alpha =1$ and $\beta ={\beta }_j$ are Gaussian distributed with mean 1 and standard deviation 0.01. Delay ${\tau }_j=5\mathrm{ms}$ and the other parameters as in Fig. 1.

Figure 4

Impact of the oscillation frequency on the coupling-induced spatiotemporal patterns in the HH ensemble (1). (a) Zigzag patterns induced by the synaptic weights $K_j$ [(b), red diamonds] when applied to in-phase synchronized reference patterns oscillating at different frequencies $f$ for homogeneous coupling $K_j=5$ and delay $\tau =20\mathrm{ms}$, as indicated in the legend. The theoretically predefined pattern is depicted by filled red diamonds and scaled by a factor of 0.25. For simultaneously modified synaptic weights $K_j$ [(b), red diamonds] and delays ${\tau }_j$ [(b), black squares, upper horizontal axis] the in-phase synchronization in the neural ensemble can be restored [(a), black squares]. The other parameters as in Fig. 1.