Planar wave approximation

In the following, we employ the results for the one-dimensional chain to give an estimate for the propagation speed of the two-dimensional system. The idea is based on the assumption that the front of the wave can be reasonably well approximated by a planar shape. The further the wavefronts that are displaced from the waves origin, the more justified becomes this approximation. For a perfectly planar wavefront, the propagation of the wave through a two-dimensional system can be mimicked by the propagation of the wave through the one-dimensional system, by rescaling the distance and coupling strength.

To see this, consider the schematic illustration of the wavefront in Fig. 3.8. Neurons shown in the same color belong to the same wavefront, in case of the left panel defined through a burst onset time within the same time bin. The displayed wave was initiated by forcing a burst in the neuron in the lower left corner, as in Fig. 3.5.

In case of a perfectly planar wavefront, as illustrated in Fig. 3.8 middle panel, neurons shown in the same color share exactly the same state(V, u). If the voltage of all horizontal neighbors is identical, links between these neurons can be discarded, because the GJ current is zero. We set the first spike time of the bursts of all red neurons as time origin. Now, every single blue neuron feels an excitatory current from two bursting neurons (connected via the links indicated in red/blue). The leak current of this one blue neuron is affected by the two links connecting this neuron to two yellow neurons (indicated by blue/yellow lines in Fig. 3.9(b)), which we again approximated to be at the resting point at this instant of time. Doubling the excitatory current and the additional leak current via GJs can be expressed by doubling the GJ conductance parameter G in Eq. (3.3.14). As can be seen in Fig. 3.8 middle and right panel, the propagation of the wave from one horizontal front

3.3 Wave propagation

q

· `

`

Figure 3.8. Wave speed in the planar wave approximation. Neural groups with simultaneous burst onset of an exemplary simulation (time resolution∆t = 0.1 seconds) are shown in the panel (a) left, for three consecutive time bins in different colors. At large distances from the origin, the shape of a wavefront can be approximated as planar, cf. (a) middle. The mechanism of burst propagation can then be mimicked by a one-dimensional situation. Therefore in our theoretical derivations, the distance and coupling strength has to be modified, cf. (a) right and details in the main text.

position to the next, does not occur along the direction of the GJ link, but the reduced distance√3/4·ℓ needs to be considered. Consequently, the velocity in the two-dimensional system can be approximated as

v2D(G) =

√ 3/4_{· ℓ} TB(2G)

. (3.3.25)

To determine the speed of a wave from simulations such as illustrated in Fig. 3.8 (right panel), we assume the wave to be circularly shaped with a fixed origin. We define a wave- front as the group of neurons that have burst onset times within the same time bin of ∆t = 0.1 seconds (exemplary groups are shown in same colors in the left panel of Fig. 3.8) and measure the front’s mean distance from the center and its mean time instance of oc- currence. The mean wave speed is calculated from the differences of these distances and times. We find the mean speed of the wave to be weakly distance dependent, cf Fig. 3.9(a), an effect that saturates at about 350 µm from the origin of the wave, which was here the lower left corner of the network shown in the inset. Accordingly, all wave speed values are averaged over measurements for the range of distances 350_{− 650 µm. This range of} distances is illustrated in Fig. 3.9(a) by the gray shaded area.

Velocities obtained by this concentric measurement method are shown in Fig. 3.9(b) as a function of the GJ parameter G. The shaded areas in Fig. 3.9(b) indicate the range of the physiologically relevant GJ conductance G _{∈ [0.1, 0.5] on the one hand and the} experimentally observed speed of stage I retinal waves,V2D=451±91 µm/sec [124], on the

other. The experimental mean value ofv2D= 451 µm/sec is attained in our simulations for

approximatelyG = 0.4. Calculated velocities v2D(G) are shown in Fig. 3.9(c) by the blue

line, underestimating the true velocity (circles) but providing a correct order-of-magnitude estimate.

3 Gap junction mediated stage I retinal waves 200 400 600 800

d

[µm]

v

[µ

m

/

sec]

G

v

[µ

m

/

sec]

Figure 3.9. Wave speed measurement and GJ dependence. Squares in (a) represent the speed of a wave with G = 0.4, measured as described in the text, as a function of the distance from the origin of the wave (lower left corner of the simulation domain, inset). The speed can also be assessed by measuring burst onset times along different fixed directions of the network, examples are illustrated as blue and green sites in the inset. The resulting wave speeds as functions of distance (blue and green lines) agree closely with the method described in the main text. Simulation results of the mean wave speed as a function of the GJ coupling G in (b). Simulation results are compared to v2D(G),

Eq. (3.3.25). The gray shaded areas indicate the physiological range ofG as derived in the methods, and the experimentally observed velocities in the rabbit retina [124], respectively.

So far, we have limited the discussion to the deterministic system,D = 0, to emphasize that the wave propagation mechanism does not rely on noise. Simulation results with noise indicate that moderate noise levels have only a very small impact on the wave speed. How- ever, retinal waves are an example of spontaneous neuronal activity during development, hence it is not a consequence of pacemaker neurons or a response to any sensory input. Much more, this activity is believed to be caused by intrinsic noise. In the following, we investigate the spontaneous nucleation of waves. Therefore we assume the neurons to be exposed to a white Gaussian noise current, which we employ to model channel noise.