Dynamical model of birdsong maintenance and control

The neuroethology of song learning, production, and maintenance in songbirds presents interesting similarities to human speech. We have developed a biophysical model of the manner in which song could be maintained in adult songbirds. This model may inform us about the human counterpart to these processes. In songbirds, signals generated in nucleus High Vocal center (HVc) follow a direct route along a premotor pathway to the robust nucleus of the archistriatum (RA) as well as an indirect route to RA through the anterior forebrain pathway (AFP): the neurons of RA are innervated from both sources. HVc expresses very sparse bursts of spikes having interspike intervals of about $2\mathrm{ms}$. The expressions of these bursts arrive at the RA with a time difference $\Delta T\approx 50\pm 10\mathrm{ms}$ between the two pathways. The observed combination of AMPA and NMDA receptors at RA projection neurons suggests that long-term potentiation and long-term depression can both be induced by spike timing plasticity through the pairing of the HVc and AFP signals. We present a dynamical model that stabilizes this synaptic plasticity through a feedback from the RA to the AFP using known connections. The stabilization occurs dynamically and is absent when the $\mathrm{RA}\to \mathrm{AFP}$ connection is removed. This requires a dynamical selection of $\Delta T$. The model does this, and $\Delta T$ lies within the observed range. Our model represents an illustration of a functional consequence of activity-dependent plasticity directly connected with neuroethological observations. Within the model the parameters of the AFP, and thus the magnitude of $\Delta T$, can also be tuned to an unstable regime. This means that destabilization might be induced by neuromodulation of the AFP.

Figure 1

Diagram of the premotor pathway, HVc and RA nuclei, and the anterior forebrain pathway, AFP, comprised of area X, DLM, and LMAN. HVc receives motor instructions which are expressed as sparse bursts to the RA and to area X. The AFP is a control and maintenance pathway. Signals from HVc through the AFP arrive at the RA with a time delay $\Delta T\approx 50\pm 10\mathrm{ms}$. The arrows at the end of lines represent excitatory couplings; the filled circles, inhibitory coupling. The dotted line is a known connection between RA and DLM whose physiological properties are not yet established.

Figure 2

The structure of the RA nucleus in our model. RA projection neurons (PNs) receive input from both HVc and, through the AFP, from the LMAN. RA-PNs are coupled with weak excitation. Populations of RA-PNs project to the syrinx and to the control of the respiratory system. RA interneurons (INs) also receive input from both HVc and LMAN. They receive excitatory signals from the PNs and project back inhibitory couplings. The arborization of the PNs is broad, and it is estimated that the ratio of PNs:INs is about 30:1 [12]. Here we represent the RA nucleus with two PNs and one IN. When there is no song input from HVc directly or via the AFP, the PNs are at rest and the INs oscillate at about $15–30\mathrm{Hz}$. The output to DLM interneurons is shown in dotted lines. The arrows at the end of lines represent excitatory couplings; the filled circles represent inhibitory coupling.

Figure 3

Membrane voltages of neurons in the model RA nucleus. Before spikes from HVc arrive at the PN and IN cells, the IN is at rest at $-61.4\mathrm{mV}$. The PNs are oscillating autonomously at $20\mathrm{Hz}$; they are weakly coupled by mutual excitation, but they do not synchronize. When a burst of five spikes from HVc arrives at $475\mathrm{ms}$, the PNs are strongly inhibited and the INs begin spiking at higher frequency. As the nucleus recovers from this input, the IN oscillations decrease in frequency. About $250\mathrm{ms}$ after the arrival of the HVc signal, the nucleus recovers completely and returns to its original state.

Figure 4

Using our biophysical model of synaptic plasticity, we evaluate the change in AMPA conductivity $\Delta g_{RA}∕g_0$ at the $\mathrm{HVc}\to \mathrm{RA}$ connections due to signals arriving from HVc followed by signals arriving from the LMAN $\Delta T$ later. In this figure the HVc signal was a burst of five spikes with interspike interval (ISI) of $2\mathrm{ms}$. A burst of five spikes with ISI $2\mathrm{ms}$ arrives from the LMAN $\Delta T$ later. The zero in $\Delta g_{RA}∕g_0$ near $\Delta T\approx 50\mathrm{ms}$ represents potentially stable plasticity in the RA, and thus a potentially stable set of connections in the song premotor pathway. Lesions of the AFP would result in $\Delta T\to \infty$ which is also a region of $\Delta g_{RA}∕g_0=0$.

Figure 5

The structure of the area X nucleus in our model. Spiny neurons (SNs) receive input from both HVc and LMAN. In turn the SN inhibits the aspiny, fast firing neurons (AFs). The AFs project inhibition to the DLM. When there is no song input from HVc, theSNs are at rest and the AFs oscillate at about $15–30\mathrm{Hz}$. The arrows at the end of lines represent excitatory couplings; the filled circles, inhibitory coupling. When the AFs are active, they inhibit DLM action. On activation the SNs inhibit the AF neurons, and with the release of $\mathrm{AF}\to \mathrm{DLM}$, the DLM neurons can rebound and fire. The $\mathrm{HVc}\to \mathrm{AF}$ connection resets the AF oscillations resulting in the $\mathrm{SN}\to \mathrm{AF}$ inhibition release of DLM becoming time coordinated with the arrival of a burst from HVc.

Figure 6

The structure of the DLM nucleus in our model. The arrows at the end of lines represent excitatory couplings; the filled circles, inhibitory coupling. The DLM-PN receives inhibitory input from the area X AF neurons. It projects excitatory processes to the LMAN. In our model, input from RA excites the DLM INs which project inhibition to the DLM-PNs.

Figure 7

The time course of membrane voltages in the AFP neurons after a burst of five spikes with ISI $2\mathrm{ms}$ arrives at SN from HVc at time $600\mathrm{ms}$. Before the HVc burst arrives at the SN, the SN is at rest near $-66\mathrm{mV}$, and AF is oscillating at $20\mathrm{Hz}$. AF activity inhibits the DLM projection neuron which shows small variations around rest with the same period as the AF. After the burst from HVc excites the SN, it inhibits the AF, and then the DLM recovers from its inhibition to fire about $67.5\mathrm{ms}$ later. The action potential in the DLM excites the LMAN.

Figure 8

Upper panel: We show the activity in the DLM-PN of the $\mathrm{Na}$ activation variable $m_{DLM}\left(t\right)$ and inactivation variable $h_{DLM}\left(t\right)$ following a burst of five spikes arriving at the SN (in area X) at time 800 ms. This indicates that the long time delay associated with the AFP is manifested in the slow recovery of the DLM-PN from its hyperpolariztion due to inhibition from the AF neuron in area X. When this inhibition is released, the DLM-PN responds with a spike as the activation variable slowly rises from 0 in the neuron’s hyperpolarized state. Lower panel: We have the same time axis and show the membrane voltage in the DLM projection neuron and in the LMAN neuron innervated by the DLM-PN firing. The DLM-PN fires about $60\mathrm{ms}$ after the HVc burst arrives at the SN, and the LMAN neuron fires about $63\mathrm{ms}$ after the HVc burst innervates the SN.

Figure 9

The time $\Delta T$ for a signal to traverse the AFP depends on the strength of the $\mathrm{AF}\to \mathrm{DLM}$ inhibitory connection. The inhibitory conductance relative to a baseline value is the ratio $R=g_{AF-DLMPN}∕g_{AF-DLMPN-\mathit{Baseline}}$. Positive slope in $\Delta T\left(R\right)$ is associated with stability in RA plasticity by the argument given in the text.

Figure 10

The AFP output when the synaptic current between the area X output neuron, AF, is changed from inhibitory to excitatory by making $V_{REVI\text{−}DLM}=0\mathrm{mV}$ instead of $-75\mathrm{mV}$. There is a burst of spikes from HVc at $980\mathrm{ms}$, but the autonomous firing of the SN and other neurons obscures the identification of $\Delta T$. The RA neuron receives many inputs from the PN in the LMAN which are not associated with an HVc burst because of the oscillations of the AFP loop. In this calculation $R=2$.

Figure 11

With $R=4$ we start the coupled HVc,RA,AFP dynamical system at $g_{RA}\left(0\right)=0.21$ and then $g_{RA}\left(0\right)=0.28$. Each initial condition lies within the same basin of attraction of the map $g_{RA}\left(N\right)\to g_{RA}(N+1)$, determined by presenting many bursts from HVc separated by $2000\mathrm{ms}$. We see $g_{RA}(N\to \infty )\approx 0.095$ and by examining the stable system we find $\Delta T=51.67\mathrm{ms}$. If we turn off the $\mathrm{RA}\to \mathrm{DLM}$ connection, the map is unstable and $g_{RA}\left(N\right)$ grows without bound.

Figure 12

With $R=10$ we start the coupled HVc,RA,AFP dynamical system at $g_{RA}\left(0\right)=0.3$ and then $g_{RA}\left(0\right)=0.4$. Each initial condition lies within the same basin of attraction of the map $g_{RA}\left(N\right)\to g_{RA}(N+1)$, determined by presenting many bursts from HVc separated by $2000\mathrm{ms}$. We see $g_{RA}(N\to \infty )\approx 0.038$ and by examining the stable system we find $\Delta T=51.14\mathrm{ms}$. If we turn off the $\mathrm{RA}\to \mathrm{DLM}$ connection, the map is unstable and $g_{RA}\left(N\right)$ grows without bound.

Figure 13

With $R=0.2$ we start the coupled HVc,RA,AFP dynamical system at $g_{RA}\left(0\right)=0.21$. In this case the map $g_{RA}\left(N\right)\to g_{RA}(N+1)$, determined by presenting many bursts from HVc separated by $2000\mathrm{ms}$ drives $g_{RA}\left(N\right)$ to smaller and smaller values. We have had to manually cut off the decrease of $g_{RA}\left(N\right)$ by imposing a lower bound of 0. This is not built into the model, but could easily be added [19]. This behavior is seen for all values of $g_{RA}\left(0\right)$ we examined for $R_{IE}=0.2$, and our qualitative arguments in the text suggest this should be so.