Figure 1

Schematic anatomical diagram of a sagittal cross section of one hemisphere of a zebra finch brain, showing the major nuclei associated with the learning and production of song. The nucleus HVC plays a central role since it receives auditory and other input, it drives the premotor nucleus RA [4], and it sends information to a parallel anterior forebrain pathway (nuclei $X$, DLM, and LMAN) that plays a role in the learning and maintenance of a song. There are two HVC nuclei, one on the left and one on the right side of the songbird’s brain.Reuse & Permissions

Figure 2

(a) Schematic diagram of a feedforward synfire chain consisting of successive pools of neurons (vertical column of circles). Each pool contains excitatory ${\mathrm{HVC}}_{\mathrm{RA}}$ neurons that fire approximately synchronously to activate neurons in the next pool. The horizontal hollow arrow on the left represents input that can initiate the synfire chain. The hollow arrows at the top of each pool indicate efferents that could convey information from a given pool to other brain areas. (b) Schematic diagram of the one-dimensional homogeneous feedforward excitatory chain that we study in this paper. In the biologically unrealistic but theoretically convenient case that all neurons in the synfire chain of (a) are identical, that all neurons in one pool connect to neurons in the next pool in the same way with identical synapses, that axonal delays between pools can be ignored, and that there is no noise, the 1D chain will have dynamics identical to a synfire chain whose spikes have become fully synchronized.Reuse & Permissions

Figure 3

(Color online) Propagation of an initial five-spike burst through an homogeneous one-dimensional excitatory chain of HH neurons, for different maximum synaptic conductances $g_s$ of Eq. (5). Each column corresponds to a fixed $g_s$ value, and successive rows of a given column show the membrane potential $v^k\left(t\right)$ as a function of time for successive neurons in the chain $(k=1,2,\dots )$. (a) For a weak synaptic coupling strength $g_s=0.1\mathrm{nS}$, the initial burst decays by the third neuron. (b) For a stronger coupling $g_s=0.2\mathrm{nS}$, the initial bursts decays to an invariant single spike. (c) For $g_s=0.3\mathrm{nS}$, the initial burst evolves to a stable state with three unevenly spaced spikes. (d) For still stronger couplings $g_s\ge 0.45\mathrm{nS}$, the initial burst can be unstable for some initial states and the number of spikes increases steadily.Reuse & Permissions

Figure 4

(Color online) The dependence of transient dynamics and resulting attractors of propagating bursts on the initial injecting current magnitude $I_e^1\left(t\right)$ and on the synaptic conductance $g_s$. For weak couplings with $g_s<0.2\mathrm{nS}$, all initial states decay to zero (not shown here). (a) For $g_s=0.3\mathrm{nS}$, there are multiple basins of attraction corresponding to asymptotic propagating bursts with one, two, or three spikes. Transients decay rapidly by the third neuron of the chain. (b) Increasing the synaptic strength to $g_s=0.35\mathrm{nS}$ increases the number of attractors, with final bursts containing one to five spikes. (c) For a larger synaptic conductance $g_s=0.45\mathrm{nS}$, initial conditions lead to stable or unstable states. An initial burst with six spikes will evolve to a slightly different invariant burst with also six spikes (squares).Reuse & Permissions

Figure 5

(Color online) Different synaptic conductances $g_s$ lead to different final propagating burst widths and spike numbers, which shows that the total burst duration is another important characteristic of the final attractor. For $g_s=0.3\mathrm{nS}$, the curve of block symbols evolves into the point $E1$. For $g_s=0.35\mathrm{nS}$, the curve with diamond symbols ends in the point $E2$. For $g_s=0.45\mathrm{nS}$, the propagation is unstable (upper triangle) except for the initial condition at point $B2$.Reuse & Permissions

Figure 6

(Color online) Plot of the number of spikes observed in a final nontransient propagating burst for different initial amplitudes of a square current pulse of fixed duration $20\mathrm{ms}$ and for different maximum synaptic conductances $g_s$. Different basins of attraction are observed, with zero-spike and unstable bursts having the largest basins and roughly equal size basins for bursts with one to five spikes in the final burst.Reuse & Permissions