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Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, Israel
Submitted 11 June 2006; accepted in final form 28 February 2007
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Chervin et al. 1988; Darbon et al. 2002; Eytan and Marom 2006; Golomb and Amitai 1997; Haas and Jefferys 1984; Miles et al. 1988; Pinto et al. 2005) to slowly evolving spontaneous activity (Sanchez-Vives and McCormick 2000; Wu et al. 1999). The investigation of such propagation is an effective and instructive method of studying neuronal networks, since it reveals both the temporal dynamics of the single neuron and the interactions between neurons that determine the network connectivity (Feinerman et al. 2005; Haas and Jefferys 1984; Pinto et al. 2005).
The propagation velocities of fronts that develop in neural cultures and slices often show high variability both across different preparations and within the same preparation (Bolea et al. 2006; Haas and Jefferys 1984; Pinto et al. 2005). For example, Bolea et al. (2006) show that the velocity of propagation for activity in a single rat hippocampal slice may vary by more than an order of magnitude along its propagation path, with high intertrial variability. The mechanisms that underlie this variability are not completely known and are subject to research (Beggs and Plenz 2003, 2004; Pinto et al. 2005). Inhibition certainly plays an important role, but is not the sole determinant because not all of the variability is reduced with blocking of inhibitory synapses (Golomb and Amitai 1997; Miles et al. 1988; Pinto et al. 2005).
We recently showed (Feinerman et al. 2005) that one-dimensional cultures grown along lines (Maeda et al. 1995; Segev et al. 2002) are a good system in which to follow propagation of fronts. Because of the linear connectivity, fronts follow a single path along which causality and order of firing can easily be identified. This avoids complications inherent in two-dimensional cultures, where many paths for propagation can coexist and where, furthermore, curvature perturbations of the front are unstable and grow to break up the front in a manner that makes it practically impossible to track (Bolea et al. 2006; Kistler 2000; Kistler et al. 1998; Wu et al. 1999).
The simplicity of one-dimensional cultures was previously exploited in several models of front propagation in neural networks. These models show that the neurons may support stable fronts (Bressloff 1999, 2000; Ermentrout 1998; Golomb and Ermentrout 2002; Kistler 2000; Osan and Ermentrout 2002; Osan et al. 2004). Synfire chain models describe propagation of activity through feedforward networks and predict that synchronization of the activity is required for stable propagation (Abeles 1991; Arndt et al. 1995; Gewaltig et al. 2001), as well as a minimal neuronal density (Diesmann et al. 1999). In these models, however, the neurons usually support only a few discrete stable propagation modes with a given amplitude and velocity. Special conditions such as a strong background noise (van Rossum et al. 2002) or fine-tuning of the excitatory–inhibitory balance (Litvak et al. 2003; Shadlen and Newsome 1998) are required to allow fronts with a range of amplitudes to propagate.
In this work we overlay the one-dimensional culture on multielectrode arrays (MEAs), enhancing the time resolution sufficiently to allow detection of weak fronts and the determination of a measure of synchrony. We show that a high variability of front propagation velocity exists in these cultures and find a link between the front's amplitude, its propagation velocity, and its synchrony. We also present a simple model based on synaptic transmission to explain these observations.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Patterned neuron cultures were grown on lines that are 70–150 µm wide and 10–45 mm long, on MEAs (MultiChannelSystems, Reutlingen, Germany). The pattern was aligned on the electrodes, so that each electrode records from a segment of the line (Fig. 1A). The surface of the MEA was prepared for neuron plating based on the techniques described in Feinerman and Moses (2003) with some modifications.
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In the second stage a protein-repelling coating was applied to the area complementary to the culture pattern. First, a metallic mask shaped to the desired pattern was aligned to cover the MEA electrodes, and thin layers composed of chrome followed by gold (10 and 50 Å, respectively) were evaporated onto the unmasked area. Next, the mask was removed and a glass tube (15 mm high, 20 mm ID) was glued to the MEA to form a chamber for containing liquids. A hydrophobic layer was created on the metal layer by filling the chamber with octadecanethiol solution (Sigma, 0.1% in ethanol, 2 h), followed by washing with ethanol and drying in air. Then a triblock PEO-PPO-PEO copolymer solution was applied (F107 Prill, Pluronics BASF, 2.8 g in 100 ml DDW, 1 h). The copolymer adheres to the exposed thiol group and creates a protein-repellant surface (Amiji and Park 1992).
The last stage began with sterilization by UV, after which uncoated areas were treated with adhesive proteins (laminin and fibronectin, 25 µg/ml each mixed with the same copolymer solution; Sigma) and left overnight (>9 h). The use of the copolymer solution in this stage was found to promote cell adhesion to the adhesive areas, presumably by competing with the adhesive proteins over the protein-repelling areas and thus further reducing their binding to these areas.
Before plating, the MEA was washed twice with D-PBS, and plating medium was applied to the MEA for 2 h. On plating, cells that were seeded over the patterned area adhere and form a network, whereas those that landed over the complementary area were rejected and either migrated to the adhesive areas or were washed away.
After the experiment the MEAs were reused by rinsing with DDW, cleansing with a neutral detergent (Sodosil RM 02, Sigma, 4% in DDW, 24 h), and sterilizing (in boiling water, 20 min).
Culture preparation and maintenance
We used Wistar rat embryos from the 19th day of pregnancy. Animals were supplied by the Animal Breeding Center of The Weizmann Institute of Science. Animals were handled according to the regulations formulated by IACUC (Institutional Animal Care and Use Committee). Dissection was done and cells were plated according to Segal and Manor (1992) at a density of 6,400 cells/mm2 in 20-mm-diameter wells, previously incubated for 1 h with 3-ml cell maintenance medium. The medium consists of 5% fetal calf serum (FCS, Sigma), and 5% heat-inactivated horse serum (HIHS, Sigma) in Eagle's Minimum Essential Medium (MEM, Sigma) enriched with 0.6% glucose, 2 mM glutamine, and 15 mg/ml gentamicin (MEM +3g). On plating, the medium was enriched with B-27 supplement. In day in vitro (DIV) 3 and DIV 4 half of the medium was replaced with MEM +3g, 5% FCS, 5% HIHS, with 20 mg/ml FUDR and 50 mg/ml uridine (both from Sigma). From DIV 6 and on, one third of the medium was replaced with MEM +3g with 10% HIHS, three to five times a week.
Maintenance of the culture while performing measurements over long periods requires minimization of water loss from the cell medium. Measurements were made at 37°C in a dry 5% CO2 atmosphere. Water loss was limited by covering the MEA chamber with a 12 µm FEP Teflon foil (GoodFellow), following Potter and DeMarse (2001).
Culture width and cell density
Pictures of the living cultures were taken using a Zeiss Axiovert 25 inverted microscope with x10 magnification. Note that fixation was not done because it may limit MEA reuse. Culture widths were measured at intervals of 500 µm along the line and then averaged. Average cell density was calculated for each culture by counting cells in four separate portions of the culture, each of them 500 µm long, and then averaging. The local density was calculated by multiplying the local width by the average density. The number of neurons per unit area is relatively constant and is 1,730 ± 290 cells/mm2 (mean ± SD). The corresponding average linear cell density ranges from 0.12 to 0.26 cells/µm.
Spike detection and sorting
The electrode voltage was amplified x1,000, low-pass filtered with a 3-kHz cutoff (MEA1060 amplifier, MultiChannelSystems), then sampled at 20 kHz, using a general-purpose DAC (PCI-6071E, National Instruments, Austin, TX). Data was acquired with LabVIEW (National Instruments) and processed using Matlab (The MathWorks, Natick, MA). The condition for spike detection was that the absolute value of the sampled signal exceeds the threshold for 
0.2 ms. The threshold was set at the maximum of 15 µV and three times the signal SD (which is normally about 3.5 µV). By spike shape discrimination, 2.5 ± 0.2 distinct neurons are identified per electrode (range is 1–4).
Front detection and characterization
The measurements for each culture were performed over times ranging between 1 and 5 h, from which extracts lasting 10 to 20 min were analyzed. All cultures showed activity in bursts that propagate along the line. To define a burst, we binned the total number of spikes in all electrodes into 20-ms bins. Candidates for bursts were groups that include one or more consecutive bins with a higher-than-average number of spikes. If the total number of spikes in this group was larger than S = 11, or if more than E = 3 electrodes participated in the activity, then this group was defined as a burst. The parameters S and E are designed to enhance detection of weak bursts because the average number of spikes in a burst was 500 ± 250 (mean ± SD) and included spikes from an average of 9.0 ± 3.3 (mean ± SD) electrodes.
Front initiation and endpoints were manually analyzed by first identifying the longest section of the line along which the front propagates at a constant velocity. A front is defined by an initiation time and position (ti, xi) and a termination time and position (tf, xf). The relevant electrodes {Ek} for identifying a front are located at positions {xk} with all the xk between xi and xf. Fronts that propagate <5 mm are ignored.
Manual analysis was verified by a computer-optimized alignment of the fronts onto the detected spikes. The manually determined front velocity is defined by c = (xf – xi)/(tf – ti). The expected time of arrival for the front at electrode Ek is 
k = ti + (xk – xi)/c. For each electrode Ek, we define 
tk as the time difference between the front arrival and the spike that is closest to the front arrival time k. The group of electrodes {Ek} constitutes those in which some spike was detected within a time t = 20 ms of the front, on the order of the membrane time constant (
m = 20 ms; Magee and Cook 2000). The alignment is optimized by varying ti and tf around the values obtained manually, in a range of ±20 ms, to minimize the jitter 
, which is defined by
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By measuring propagation along a unidimensional path we maintain causality, so that the sequence of electrode positions along the path reliably portrays the sequence of excitation of the neurons. We used the high temporal resolution of the MEA and relied on an assumption of a constant propagation velocity, which we verified independently, to detect fronts even if they generate spikes in only some of the electrodes. The lower limit on the amplitude that can be detected is about 10 spikes · s–1 · electrode–1. Taking into account that we measure within a 30-ms window and record about 2.5 neurons per electrode, we are sensitive even to fronts in which only 13% of the neurons spike.
Front synchrony
We estimated the level of synchrony of a front by observing the onset of activity along the front's propagation path. The synchrony of the front was defined by firing of neurons within a single synapse time from the front's arrival. For the synapse time tsyn we took the excitatory postsynaptic potential (EPSP) peaking time tsyn = 3 ms (Fricker and Miles 2000; Magee and Cook 2000). Synchrony occurs when the spike measured on an electrode occurs within tsyn of the front's arrival. We therefore defined a "measured" synchrony index of a front by
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The measured synchrony index (SIm) is partially determined by the amplitude; the more spikes that the neuron fires, the higher is the probability that one spike will randomly fall in the first 3 ms. We therefore calculated the "predicted" synchrony index (SIp) that is given solely by the amplitude and the associated temporal spike distribution.
Calculation of SIp used the measured distributions from experimental data. We defined S(t) as the probability to spike as a function of time relative to the front arrival time and P(n) as the probability to detect n spikes within 20 ms of the front arrival time. We used 1,774 fronts from nine cultures, of which 1,601 were spontaneously generated, 60 chemically evoked, and 113 electrically evoked. The full velocity range (0–410 mm/s) was split into 10 working ranges and the distributions S(t) and P(n) were measured for each velocity range. We simulated the front arrival to M electrodes by generating random spike trains that include a number of spikes according to P(n) with a temporal firing distribution according to S(t). By using M = 106 we limit the error in SIp to about 0.1%. SIp was then calculated by taking ES as the number of electrodes that fired within tsyn after front arrival and ER as the rest of the electrodes, provided that they fired at least once within 20 ms.
We finally defined the synchrony index as the difference between measured and predicted SI values
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Stimulation of the culture
In this work we used two methods of stimulation. Electrical stimulation of n = 5 cultures was achieved by a 500-µs, 280- to 570-mV positive leading bipolar pulse at one of the electrodes (Wagenaar et al. 2004). We did not record from the electrode used for stimulation. We excited in two to four different locations along each culture. The threshold voltage required to evoke a propagating front was 200–500 mV, depending on the stimulation site. Below the threshold, neurons in electrodes closest to the stimulation site could be stimulated, but a propagating front did not emerge. To limit the effect of the stimulation on the culture, stimulation sessions were limited to 3 min in each location. Chemical stimulation of n = 2 cultures was achieved using local application of 100–200 µM glutamate applied with a concentric double pipette (Feinerman and Moses 2003; Feinerman et al. 2005). The drug is confined to a region of 100-µm diameter, by collecting the drug after it affects the target neurons and before it diffuses to the surrounding medium.
For both methods the stimulation rates (0.125–0.2 Hz) were chosen to be lower than the spontaneous activity rate by 50% to keep the culture responsive and to minimize changes in the network (Eytan et al. 2003; Maeda et al. 1995).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Spontaneous front initiation and propagation
We measured the neural activity during the third week in culture (DIV 13–20). The average culture burst rate was 0.4 Hz, ranging between 0.15 and 0.75 Hz, and the average spiking rate in the burst was 3–10 Hz per electrode. Roughly 95% of the spikes were within bursts and the culture was supporting a burst during 20 to 80% of the recorded time interval.
The bursts initiated at a point along the one-dimensional culture and propagated along the line. The average front propagation length was 9.1 ± 4.5 mm (mean ± SD). The fronts usually terminated either at a weakly connected part of the culture, or at one of its ends. When the culture had weakly connected parts, the bursts either terminated at the weak connection or passed through with a significant delay, as described in Feinerman et al. (2005). The geometry of the culture and the activity patterns originating from spontaneous activity are exemplified in Fig. 1, A–D.
We checked whether the velocity of the front is determined by its initiation point by comparing the velocities of fronts that are initiated in each third of the culture relative to those initiated in the whole culture. In most cultures (seven of ten) there was no significant relation between the front velocity and its initiation point (P > 0.05 using Student's t-test).
In 77% of the bursts a well-defined front propagating with a constant velocity was identified. In the other cases, the front was not well defined and was excluded from analysis. In most cultures, several locations could serve as burst initiation zones and usually two to four points dominated.
A preferred propagation direction was observed in five of ten cultures tested (P < 0.05 comparing the numbers of fronts detected in each direction assuming they are generated by a Poisson distribution). In these cases, the dominant initiation point was located at one edge of the culture. Different velocities (10–30% difference) were measured for front propagation in opposite directions for five of the ten cultures (not all the same five cultures as mentioned earlier).
Stability of the front velocity and amplitude
To verify our assumption that the velocity is constant despite small inhomogeneities in cell density, we tested n = 345 fronts from two different cultures, repeating the analysis described earlier in Front detection and characterization, taking the first spike rather than the closest to the front and using a wider time window of [–100, 100] ms. This gave a linear relation between the spike time and the electrode position along the line with an average correlation of R = 0.92 ± 0.01 (mean ± SE is presented here or below unless noted otherwise) for all the fronts. For 95% of the fronts, the linear fit resulted in significance of P < 0.05 (unless otherwise stated, throughout we use Pearson's standard correlation). We tried to improve the fit by using a quadratic function but this did not result in an improvement of the fit. We therefore conclude that the fronts do indeed propagate at a constant velocity along the culture.
We checked whether amplitude is conserved along the line by comparing between the front amplitudes measured in either the first or the second half of the path to the average amplitude over the full path. Only fronts with propagation length of 10 mm were considered and amplitudes were averaged over a period of [–10, 100] ms relative to front arrival time. The correlation coefficient between half and full lengths was R = 0.89 (P < 10–16), which implies that amplitudes are conserved along the front propagation path.
High variability of spontaneously generated fronts
To characterize the distribution and its width we use the coefficient of variation (CV), defined as the ratio between the SD of the sample and its mean. We calculate the CV value for the spontaneously generated fronts for each culture separately. Average CV values for front velocity and amplitude are 0.72 ± 0.10 and 0.66 ± 0.05 (n = 10), respectively. Such high values of CV indicate rather broad distributions.
A possibility that must be ruled out is that the front velocity is affected by the history of firing. Preceding bursts alter the network state and front variability could result from the different resting times that passed between consecutive bursts (Darbon et al. 2002; Maeda et al. 1995; Yvon et al. 2005). However, we found no significant correlation between the front amplitude and the time from last burst initiation for eight of the ten cultures tested. In two cultures a correlation (with R = 0.19 ± 0.05, P = 0.06) was found. It is interesting to note that under the culture maintenance protocol used in Feinerman and Moses (2006), the correlation between amplitude and interburst interval was significant (n = 3, R = 0.47 ± 0.06; mean ± SD), while they reported a weak correlation.
Effect of cell density
The average culture widths ranged between 70 and 150 µm and the cell densities ranged between 0.12 and 0.26 cells/µm. The density of each culture was reasonably uniform, with the SD of the local density equal to 25% of the average. Inhomogeneities occurred in some of the cultures with two extreme forms, either in clustering of neurons or in the creation of barely connected parts (Fig. 1A). Overall, >90% of the culture was free from clusters or disconnections.
The effect of such inhomogeneities was local and specific. Clusters are preferred locations for spontaneous generation of fronts. Barely connected parts cause either delay or complete halting of the front. In such cases, which rarely occur, we restricted our analysis to the homogeneous propagation parts of the front.
The mean velocities, as averaged over all fronts within each culture, are linearly related to the width and thus to the average cell density of this culture (R = 0.90 with P < 0.001; Fig. 2). High correlation also exists between the average front amplitude of each culture and the average cell density of the culture (R = 0.84 with P < 0.005).
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Relation of front amplitude and velocity
As mentioned earlier, the spontaneous activity we measure exhibits a wide spectrum of front velocities and front amplitudes that spans about an order of magnitude for each of the cultures. For these fronts, the velocities and amplitudes are correlated, as can be seen in Fig. 3A. The amplitude versus velocity obeys a linear relation, with a slope of 0.75 ± 0.03 spikes · mm–1 · electrode–1, with R = 0.98 and P = 2 x 10–7.
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Effect of stimulation on characteristics of the fronts
Electrical stimulation was applied to n = 5 cultures, resulting in propagating fronts with velocities ranging between 354 and 405 mm/s. Amplitudes were measured to be in the range of 111–187 spikes · s–1 · electrode–1. In all the samples the propagation velocities were faster than the averages for spontaneous activity by a factor of 3.1 to 3.3. The amplitudes were higher by a factor that varied between samples and ranged from 0.8 to 2.0, with an average of 1.3 ± 0.1 (Fig. 3).
It is instructive to note that the electrically evoked fronts deviate from the linear relation of amplitude and velocity derived for spontaneous activity and show signs of saturation. This may indicate that a difference exists between the electrical and spontaneous activity. This is also supported by the synchrony we measure (see Front synchrony section below). Saturation is unexpected in this amplitude range because amplitudes 
500 spikes · s–1 · electrode–1 seem to be measurable.
Local chemical stimulation (on n = 2 cultures) resulted in front velocities of 96–154 mm/s, within 20% of the average velocities of spontaneously generated fronts. The amplitudes are in the range of 70–95 spikes · s–1 · electrode–1, similar to the amplitudes of spontaneous activity in the same velocity range (Fig. 3). This stimulation method seems to evoke fronts that resemble spontaneous activity fronts.
For both stimulation methods, the variability in velocity was lower than that measured in spontaneously generated fronts. The values obtained for the velocity were CV = 0.63 (n = 247) for spontaneous activity, CV = 0.17 (n = 157) for electrically evoked fronts, and CV = 0.35 (n = 65) for chemically evoked fronts. The CV decrease is significant, in a comparison of the spontaneous to either stimulation method, giving P < 10–16 for electrical stimulation and P < 3 x 10–7 for chemical stimulation, using Levene's test for homogeneity of variance on the velocities normalized to their average. The variability of the amplitude was also significantly lower, especially for electrical stimulation: spontaneous, CV = 0.63; electrically stimulated, CV = 0.163; P < 10–16; glutamate stimulated, CV = 0.50; P < 10–8.
The effect of inhibitory–excitatory balance on spontaneous activity
To evaluate the role of inhibitory neurons on the variation in spontaneous fronts, we applied a treatment that includes 20 µM of the 
-aminobutyric acid type A (GABAA) antagonist bicuculline and 20 µM of the N-methyl-D-aspartate (NMDA) antagonist 2-amino-5-phosphonovaleric acid (APV) to n = 4 cultures. This treatment leaves the 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) synapses dominant in the culture activity. Figure 4 depicts the average response and includes 443 fronts from untreated cultures and 178 fronts from treated cultures. In all the cultures the disinhibition increased the amplitude and eliminated the low-velocity part of the front spectrum, thus strongly reducing the amplitude and the velocity variability. This is seen from the reduction in both the CV of the propagation velocity (from 0.47 ± 0.04 to 0.22 ± 0.03) and in the CV of the amplitude (from 0.62 ± 0.03 to 0.26 ± 0.02). The average front velocity increased from 175 ± 6.5 to 269 ± 6 mm/s. The slope of the linear relation between amplitude and velocity for the drug-free cultures was 0.88 ± 0.04 and, when treated, it was 2.70 ± 0.65, increasing by a factor of about 3.
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Front synchrony
We use the index of synchronization SI (see METHODS), based on the fraction of electrodes with onset of spiking earlier than the synapse time tsyn = 3 ms. We calculated the SIp values based on measured amplitude and temporal spike distribution in 10 velocity ranges (see METHODS). The synchrony index (SI) as well as the measured and predicted synchrony values (SIm, SIp) are depicted in Fig. 5. Beyond the fact that the synchrony index SI is positive for all velocities, and significantly positive for velocities >100 mm/s, a number of other observations stand out. SI of the spontaneously generated fronts increases with the front velocity and therefore also with amplitude, giving a relation of SIspont = av + b, where v is the front velocity in mm/s, a = 5.4 ± 1.1 x 10–4 s/mm, and b = 0.17 ± 0.02.
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Comparison to previous work on one-dimensional cultures
Previous calcium fluorescence imaging measurements of unidimensional cultures (Feinerman and Moses 2006; Feinerman et al. 2005) differed in three of our major findings. They did not report a significant variability in front amplitude or in velocity and the mean velocity was lower. Specifically, the mean amplitude varied from front to front by no more than 10–20% and the mean velocity was reduced by a factor of 3 for the same cell density.
The main difference between the present and previous work lies in the culture maintenance protocol. In the present work, the feeding schedule includes a partial change of the culture medium three to five times a week. In the previous work the culture medium was not replaced from DIV 6 until the day of measurement (DIV 14–21), when the cells were transferred to a recording solution. To test whether the feeding protocol alone caused this difference in behavior, we grew n = 3 cultures using the culture maintenance protocol of Feinerman et al. (2005) on MEA, keeping them in a humid atmosphere to minimize water loss. On the day of measurement (DIV 14), we exchanged the growing medium to the recording medium.
The average velocity measured on MEA with this protocol was 54 ± 7 mm/s, similar to the previously published value. The bursting rate of 0.05 ± 0.03 Hz was also similar to the rate of 0.05 ± 0.01 Hz seen in the experiments of Feinerman and Moses (2006). The velocity distribution had CV = 0.43 ± 0.10, which matches the value of CV = 0.46 ± 0.08 that can be obtained from the data of Feinerman and Moses (2006).
Although the velocity issue is resolved by these controls, the amplitude distribution width was CV = 0.54 ± 0.17, higher than the value of CV = 0.14 ± 0.04 obtained from the data of Feinerman and Moses (2006).
A second reason for differences in the results lies in different analysis algorithms for the data. To control for this we performed on our MEA data a secondary analysis that corresponds to the one given previously to the fluorescence data. The relevant limitation set by fluorescence on the analysis is the need for averaging over several neurons and over a time window set by the escape rate of Ca from the cell (on the order of 1 s). We limited the analysis to cultures with comparable cell densities and to fronts that were not preceded by any activity for 500 ms and that propagated >10 mm. The amplitudes were calculated by averaging over a window of [–50, 100] ms.
Under these conditions, the measured spread of the amplitudes resembles those measured previously (Feinerman et al. 2005), with values CV = 0.2–0.45.
It is interesting to reanalyze the previous experimental data and search for the amplitude versus velocity relation. Figure 6 shows the existence of a linear relation between the amplitude and the velocity (R = 0.95 and P < 0.005 for n = 7) in the previously measured calcium fluorescence data from Feinerman and Moses (2006). This relation was not detected previously because of the relatively weak variation in amplitude.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Haas and Jefferys 1984; Pinto et al. 2005), with narrower velocity distributions in disinhibited preparations (Golomb and Amitai 1997; Miles et al. 1988). Variability of front amplitude and velocity
The variability in velocity that we measured for spontaneous activity is in the higher range of results in the literature (Bolea et al. 2006; Haas and Jefferys 1984; Miles et al. 1988). The results for slices have less variability (CV = 0.1–0.3), which may be explained by our observation of the reduction in variability caused by disinhibition or by electrical stimulation, both of which are used to evoke tractable fronts in slices (Golomb and Amitai 1997; Miles et al. 1988; Pinto et al. 2005). In our experiment these caused the variability to drop to CV 0.3, which is comparable to the slice measurements. In other cases the reported velocity variability is similar to ours, with fronts propagating over 1–5 mm at a velocity range (in mm/s) that spans 40–150 (Haas and Jefferys 1984), 3–60, and 98–167 (Bolea et al. 2006). The variability in velocity was not previously related to variability in amplitude.
What then is the origin of the variability in front amplitude and velocity that we measure? Because the same neuronal substrate supports a multitude of velocities, we tend to ascribe the origin of this diversity to the variability of the initial conditions that characterize the firing of the culture. It seems that spontaneous activity can be generated with a multiplicity of initial amplitudes. This can lead to different synchronies of the front and therefore to different velocities. The electrically and chemically stimulated activity differs from the spontaneously generated ones and is characterized by more uniform fronts.
Golomb and Ermentrout (2002) suggest a role for inhibition in creating variability in velocity of the front, although they do not indicate what is the precise role of inhibition. Our results are consistent with this because disinhibition indeed dramatically reduced the variability.
Effect of cell density
Both the average velocity and the average amplitude in spontaneously generated fronts are proportional to the average cell density (Fig. 2). This relation may be explained in two ways. First, when the front propagates, EPSPs arrive at neurons that have not yet fired. The EPSP arrival rate is linear with the presynaptic neuron density. Assuming the same axon length for all neuron densities, the neuron recruitment into the active front will be slower for a thinner culture and the maximal spiking rate will be lower, leading to lower front velocity and amplitude. Below some critical linear neuron density, we expect fronts to halt because EPSP rates below that critical value will not cause downstream neurons to fire. Although some compensation could come from the fact that the synaptic strength is expected to be stronger for lower-density cultures (Kirov and Harris 1999) this compensation seems to be only partial.
A second possible explanation is that higher-density cultures, which are also wider, allow the growth of neuron processes to a longer length that enables faster propagation velocity. Although we do not reach a conclusion as to which alternative is correct, we tend to believe the first argument is more convincing. Plating of cultures with different cell densities but with similar widths could resolve this question, but we met with difficulties in obtaining viable cultures with very low or very high densities.
It is interesting to deduce from the x-intercept of Fig. 2 a lower bound on the density below which a front cannot propagate. This results in a value of 0.13 ± 0.06 cells/µm, which implies culture width of 48 ± 22 µm.
Model for the amplitude–velocity dependency
Here we derive a model that explains the linear relation between amplitude and velocity and shows the role of synchrony in the propagation of fast fronts. We consider the process of recruitment by which the front is propagating, follow the general reasoning suggested by Feinerman et al. (2005), and take into account the firing rate amplitude. We assume that the first spike to reach a neuron at position x0 arrives at time t = 0 and denote by ts the time at which it eventually fires. This first spike is sent from a presynaptic neuron positioned at a distance of one axon length 
preceding x0 and we assume that the axonal propagation is instantaneous. During the time ts, the front will propagate a distance . This sets the front propagation velocity to be c = /ts.
The number of excitatory neurons within a distance is N = 
f, where is the SD of the axon projection length distribution along the line, f is the fraction of excitatory neurons, and is the linear density of neurons.
The postsynaptic neuron will fire if its internal voltage exceeds a threshold voltage VT. Denoting by r the individual somatic EPSP rise time and by m the membrane time constant, we consider the case in which r << ts < m. Within this timescale, a simple integration of the EPSPs can be performed and the condition for firing after time ts is
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The ratio R/c for AMPA-dominated cultures can be calculated from the experimental data and is (R/c)AMPA = 0.54 spikes · mm–1 · cell–1. Taking = 308 µm from Feinerman and Moses (2006) as the relevant length scale, f = 0.75 as the fraction of excitatory neurons in our cultures (J. Soriano-Fradera, personal communication), and = 0.25 cells/µm as the cell density per unit length for the thicker cultures, we arrive at a value of VT/gs = 4.7. This is about half the value obtained in Feinerman et al. (2005), which is a direct result of the faster velocities we measured. This goes back to the issue of different feeding protocols and a fascinating open question is which of the factors are specifically affected by the feeding.
To estimate the range of R over which Eq. 3 holds for spontaneous activity of untreated cultures, we use the values (R/c) = 0.3 spikes · mm–1 · cell–1, which we derived from our data for untreated cultures, of r = 1 ms and m = 20 ms at 37°C (Magee and Cook 2000). The smallest rate R for which this relation holds is derived from the limit ts < m, and substitution into Eq. 3 yields Rmin = 4.6 Hz. The maximal bound for R is derived from r < ts, yielding Rmax = 92 Hz. This range is consistent with that measured for spontaneous activity in our cultures, which we found to be between 10 and 160 Hz per electrode, or 4–64 Hz per neuron. We verified that refining the argument with precise calculation of the EPSP form does not significantly improve this estimate.
It is illuminating to consider the extremes at which our model begins to lose applicability. At the higher-velocity range the data deviate from the model and the amplitude is lower than expected. We understand this as the result of synchrony, in which the nonlinear bunching of spikes results in high velocities (as in the synfire model).
Stimulation of the culture
Electrical and chemical stimulation evoked fronts with reduced widths for both the velocity and the amplitude distributions, compared with spontaneously generated fronts. This is in agreement with Pinto et al. (2005) who saw a reduction in the scatter of front velocities as a function of the electrical stimulus strength. Our explanation for the lack of variability is that by stimulating the culture we force the initial conditions of the activity, which then are conserved as the front propagates along the line. This does not occur in the case of spontaneously generated activity.
The electrically evoked fronts are faster and highly synchronous, presumably because of the short timescale imposed by the stimulation. Fronts evoked by local glutamate application exhibit amplitudes and velocities that are more similar to those measured in spontaneously generated fronts. This is understood in terms of common initial conditions because Feinerman et al. (2005) found that a slow and low-amplitude recruitment stage exists for both glutamate-evoked fronts and spontaneously generated ones.
One further conclusion with possible implications for experimental design is that stimulation of the culture with chemicals yields responses that are close to the mean (which is also the most probable velocity) of naturally evoked spontaneous activity. This is not so for electrical stimulation, for which the measured velocities are substantially higher.
Comparison to previous work on unidimensional cultures
Comparison of our measurements with previously published work on unidimensional cultures (Feinerman et al. 2005) revealed clear differences in the average propagation velocities and in the velocity distribution widths, which were shown to result from the different culture maintenance protocol. The extended amplitude distribution widths measured in the present work were caused by the improved temporal and spatial resolution of the multielectrode arrays relative to calcium fluorescence imaging. We note that amplitude–velocity correlation can be detected, even with the limited amplitude variation in the calcium imaging data.
Comparison to existing models
Synfire chain models (Abeles 1991; Diesmann et al. 1999; Litvak et al. 2003; Mehring et al. 2003) regard the propagation of activity waves in feedforward networks and have been proposed as a mechanism to maintain exact timing in neural activity. Generally, they predict that propagating activity in feedforward networks tends either to synchronize within <10 synaptic hops or to die away. The velocity of propagation of a synchronized activity wave may vary because it depends not only on the synaptic strength (Arnoldi et al. 1999; Wennekers 2000) but also on the firing threshold (Arndt et al. 1995). In addition, the variation of velocity might result from a variation in the synchrony, which depends on the number of neurons per pool (Diesmann et al. 1999) and on the level of synchrony of the initial conditions (Diesmann et al. 1999; Gewaltig et al. 2001). Partially synchronized activity, which propagates slower than fully synchronized activity, was suggested to be stable in the case of inhibitory input to the network (Arnoldi and Brauer 1996). Background noise was also shown to cause waves to propagate in various velocities (van Rossum et al. 2002; Wennekers 2000; Wennekers and Palm 1996).
Several unidimensional neural media models have been presented to-date (Bressloff 1999, 2000; Ermentrout 1998; Golomb and Ermentrout 1999, 2002). In Golomb and Ermentrout (2002), for example, two stable propagation modes may exist for a single set of network parameters, which differ by an order of magnitude in their velocities. Simulations of simple systems with noise (van Rossum et al. 2002) or with inhibition (Compte et al. 2003) do indicate that for a single set of network parameters, more than one velocity may be sustained.
Our measurements indeed show that when the wave is more synchronized, it propagates faster. This was observed over the whole spontaneous activity spectrum and when the culture is disinhibited the effect is even stronger. When stimulated by roughly 1-ms stimulation, synchronization of spikes is imposed on the front and maintained along the line. Higher linear cell density is also linked to faster and more synchronized fronts emerge spontaneously.
Our results suggest that neural cultures may indeed sustain propagating activity that is partially synchronized. We regard the variability in the amplitudes and in the velocities of the fronts not as a difference in the network parameters, but rather as originating in a variability of the initial conditions of the front generation.
In conclusion, we have shown that in one-dimensional cultures, activity fronts can propagate with widely ranging velocities and firing amplitudes. The fronts propagate with velocities that are correlated with their firing amplitudes and with their synchrony, as suggested by Feinerman et al. (2005). Characteristics of fronts are controlled by their initiation. The predictions of propagating fronts models such as synfire chain models and one-dimensional continuous media models do not seem to describe our results in full. To explain the new results quantitatively, we used a simple model that takes only excitation into account. We conclude that models that take into account variable firing rates and variable synchrony as well as inhibitory–excitatory balance need to be developed.
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Address for reprint requests and other correspondence: S. Jacobi, Department of Physics of Complex Systems, The Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel (E-mail: shimshon.jacobi{at}weizmann.ac.il)
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