Fig 1.
A graphical representation of the neural network described.
The two rings are HVC and RA neurons connected with a simplified small world (S.W.) network structure of dimension one. The left ring represents a number of HVC neurons connected to their adjacent neighbors. In a similar fashion are setup the RA neurons in the adjacent ring on the right. To elucidate the HVC-RA stochastic connection algorithm, the first few HVC neuron are schematically numbered as “1”, “2”, “3” etc. and the random connections to RAs neurons are represented by lines going on the right. The probability of connection to RA grows counterclockwise (arrow) to realize the sequential spiking. Connections are unidirectional from HVC to RA and are indicated by the curved lines (not all connections are shown, those from neuron 4 onwards are interrupted for clarity, connections are random and differ for each simulation). Each neuron is a python object that executes the Izhikevich neural model (Eq 1). Half of RA neurons connect to muscles that drive the bronchial pressure (P) and half to the labial tension (T). A neuronal recruitment model is used to integrate neuronal spikes and activate muscles (Eqs 4 and 5). RA neurons connect to a physical model of the syrinx regulated in this way by tension and bronchial pressure (Eq 6).
Fig 2.
Top panel: the average membrane potential for a group of 20 HVC neurons (connected as in Fig 1) in function of time (each point is 1 msec). Bottom panel: the Fourier transformation of the same membrane potential. Strong low frequencies delta waves are prominent, peaks at higher frequencies in the α waves range are also noticeable.
Fig 3.
Distribution of the membrane potential.
Top panel: The membrane potential distribution calculated for 20 RAs cells, these cells activity is sharper and spread in a narrower range compared to the HVCs cells. Bottom panel: the same distribution for the HVCs cells group. Activity is sparse over a wider range of potential values. Parameter of the simulation are as in Fig 5.
Fig 4.
Panel (a): two notes song of a common songbird, the great tit Parus major. In panel (b) the resulting sounds from our model is shown for comparison. The sounds produced by the very simple neural network described here, resemble real birdsong in intermittent behavior, tonality, syllable duration and fundamental frequency. An example wave file representing the sounds generated by our model is in the supporting media S1 Audio: “tEvol_PLOS.wave”. Simulation parameters in this plot are those of Fig 5 and sampling rate increased of a factor of three for easier visualization and comparison.
Fig 5.
A simulation of the spiking neural network model described.
Burst of oscillations produce spikes of pressure and tension (upper panel) that consequently generate oscillation of the syrinx (lower panel). Top panel: recruitment tension and pressure are shown in red and blue, respectively. Bottom panel: bird syrinx movement in X and Y axes are shown in green and orange, respectively. The insets on the top left and top right represent a zoom-up of the first burst in XY and pressure/tension respectively. Simulation settings: C = 0.4, neurons number = 20, tau = 10 ms, INLVL = 0; I current = 10mA. The horizontal axis represents the integration time steps dT = 0.1ms, so total duration of simulation is one second.
Fig 6.
The labeled positions of relevant frequencies in the birdsong spectrum as Fourier transformed from Fig 5.
Peak numbered 1 represents the minimal frequency (“Min Freq”) at which the sound begins, number 2 (“Peak 1”) the actual main peak of the sound, number 3 (“Peak 2”) frequency at which the sound collapse, number 4 (“Peak 3”) is the second high-frequency position and number 5 (“Peak 4”) the higher frequency position that characterizes all the sounds produced by the model and represented in Figs 7 to 10. Bird syrinx movement in X and Y axes are shown in green and orange, respectively. The horizontal axis scale is 104Hz, so 0.05 corresponds to 500Hz (every time step corresponds to dT = 0.1 ms). The highest peaks 4 and 5 are in some cases not discernible and will be omitted in the plots.
Fig 7.
The spectrum characteristics of the complex sound generated by the network in function of network size.
Interestingly the central frequency peak of the birdsong increases with network size then have a plateau at about 15-20 neurons where the frequency is more than double than for lower or higher number of neurons. This is a resonant phenomenon that, we hypothesize, could be a model of spontaneous self-regulating mechanisms of higher complexity in real brains. Vertical axis is Hz*10000 and horizontal the number of neurons in the network. Experimental settings: C = 0.4, tau = 10 ms, INLVL = 0; I current = 10mA.
Fig 8.
The spectrum of the oscillation of the syrinx is studied in function of the input current of the single neuron that initiates the cascade of spiking in the network.
The current level affects the neuron spike rate and a resonant phenomenon is observable around 50 and 80 mA. We can speculate that Birdbrains may modulate the emission frequency by intervening on this current and this would explain variability of birdsong frequencies. Experimental settings: C = 0.4, neurons number = 20, tau = 10 ms, INLVL = 0.
Fig 9.
The spectrum of the sound generated by the model is a function of internal noise present in the network.
The main peak seems to remain in the order of 500-600 Hz, however central values show a resonant opposite trend (negative peaks around INLVL = 4). Noise is expressed in mV and frequency in 104Hz, as in previous plots. Experimental settings: C = 0.4, neurons number = 20, tau = 10 ms, I current = 10mA.
Fig 10.
Here we manipulate the physical parameters that describe the decay of bronchial pressure and muscle tension in the physical model of the syringeal and bronchial apparatus (Eq 4).
The model results to be robust to variation of τ, since there are no major changes in most of the range used (20 to 60 ms). This supports the idea that the labial response is driven mainly by the neural pathway and weakly influenced by the physical properties of the syringeal system. Experimental settings: C = 0.4, neurons number = 20, INLVL = 0; I current = 10mA, τp = τt for simplicity.