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Conceived and designed the experiments: CT SO UE FW MB. Performed the experiments: CT SO. Analyzed the data: CT. Wrote the paper: CT FW MB.

The authors have declared that no competing interests exist.

Recently evidence has accumulated that many neural networks exhibit self-organized criticality. In this state, activity is similar across temporal scales and this is beneficial with respect to information flow. If subcritical, activity can die out, if supercritical epileptiform patterns may occur. Little is known about how developing networks will reach and stabilize criticality. Here we monitor the development between 13 and 95 days in vitro (DIV) of cortical cell cultures (

Learning depends crucially on the synaptic distribution in a neural network. Therefore, investigating the development from which a certain distribution emerges is crucial for our understanding of network function. Morphological development is controlled by many different parameters, most importantly:

During the last years increasing evidence has accumulated that networks in the brain can exhibit “self-organized criticality”

In order to assess how self-organized criticality develops in cell cultures, we have monitored a total of 20 cultures and recorded their activity patterns between 13 and 95 days in vitro (DIV). In general, cultures start with about 500,000 dissociated cortical neurons, which develop over time into an interconnected network. To assess the different network states the activity at 59 electrodes was measured and analyzed at different DIV (see

They are showing (

At early stages during development, usually before 13 DIV, connectivity is small and activity in the network very low. So, it is very difficult to obtain long enough recordings for plotting avalanche distributions. However, known from the literature

The transitions from initial (black) to supercritical (red) to subcritical (green) and critical state (blue) can be clearly seen. Data from the same cell culture at different time points are connected. 14% of the total number of cultures has been tracked at 5 different time points, 7% at 4 time points, 29% at 3, 14% at 2, and 36% once. Squares indicate the mean values of DIV and

The dashed line indicates a perfect power law distribution. The deviation of the cell culture data from this line measures the criticality of these systems. For each state three different examples are shown. The age of each state of the cell cultures is given in the bottom right corner of the panels. (

In summary, these results show that there is a characteristic time course in the development of the avalanche distributions. The system starts with low activity and then enters a transitory initial state. Quickly it leaves this state and, passing supercritical and subcritical regimes, reaches the critical state.

The model uses two opposing mechanisms of axonal and dendritic growth and is driven by the goal to reach homeostasis of the mean firing rate. The first mechanism regulates dendritic growth probabilities inversely to neuronal activity and the second is the axonal outgrowth promoted by activity. Specific choices for the model are being discussed in the

As will be shown below, the model is capable of reproducing all different patterns of neuronal activity (

Dependent on the difference between the current calcium concentration and a desired homeostatic value

The difference between an inhibitory and excitatory neuron is defined by constants

In summary, the model comprises a negative feedback loop of the following kind (

(

These interactions lead to the effect that the model development passes through three different morphological phases (

The initial supplies of the axons and acceptances of the dendrites are chosen such that no connections exist. As a consequence of the resulting too low activity the dendritic acceptance increases to build synapses and to enhance the activity in the first developmental phase I. It rises slowly and, at a certain point in time, increases explosively towards a maximum. Parallel to this increase in activity, the system undergoes a morphological transition (Phase II) until it reaches homeostasis (Phase III). As discussed later (see

The three different phases in the above described development can be largely understood in an analytical way and we can also describe to what degree the system approaches criticality. The difficult recurrent processes, which drive the interactions within a network and lead to a specific avalanche distribution, however, defy analytical analysis and can only be obtained from simulations. Additionally, the effects of inhibition on the network dynamic in the different developmental phases are tested by simulations.

The first phase (Phase I) of the network development is characterized by dendritic growth to establish first synaptic contacts and to rise neuronal activities. At the beginning of the model development the dendritic acceptance increases (

For the initial conditions of the model without connectivity,

Taking the limit

From

Gray areas in insets (taken from

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As soon as the system has reached small connectivity, the behavior of the membrane potential, calcium concentration, and avalanche distribution changes. This corresponds to a situation where we have

We can solve the differential equations (Equation 14 and 15) for the mean variables with standard methods and get:

Also the avalanche distribution changes slowly with rising connectivity and activity from a Poisson to a power law distribution (see transition in

In the whole Phase I, the network never attains steady state. Hence connectivity and activity continue to change. Criticality essentially follows these changes. The transition from small to medium synaptic density only leads to a qualitative change in the distribution (

Phase II of the network development is characterized by an overshoot in network activity. The membrane potential and calcium concentration (

We can calculate the membrane potential as before with Equations 1 and 2 now with the constraint

If the membrane potential is close to one, neurons theoretically fire at every time step. Due to the given refractory period of 4 time steps, however, only

Introducing inhibition changes this behavior substantially. The mean membrane potential decreases from

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As in Phase I, the network will not reach a steady state in Phase II, either. However, by contrast to the first phase where activity and connectivity is slowly growing, in the second phase, connectivity and activity is quickly getting overly strong (

Phase III is that of morphological homeostasis of the network and the network has now equilibrated reaching a steady state, where firing rate is stable in the mean.

It is obvious that the average steady state rate

Let us first consider the system without inhibition. Also in this case in Phase III we receive a stable rate with constant

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With different levels of inhibition the steady state connectivity

As a central conclusion we observe that rates

As the firing rate is the most accessible variable in cell cultures, we are now showing how to compute the firing rate in the model analytically. When the network is in a homeostatic equilibrium, the calcium concentration for each neuron on average equals the target value

Above we observed that inhibition influences the final connectivity that gives rise to network homeostasis. Here we find that also the avalanche distribution is dependent on inhibtion (

As a characteristic example the avalanche distributions from fixed point 1 (see

This demonstrates that criticality after equilibration, hence on the long run, depends on connectivity but neither on the mean membrane potential

This can be nicely demonstrated by disturbing an equilibrated system by a sudden change of inhibition (

Inhibition is suddenly decreased/increased (

A comparison between panel B in

Thus, the model predicts that sudden activity changes should affect criticality in Phase III, but in a reversible way. Lasting changes of inhibition, on the other hand, should also lead to lasting small changes in the criticality

So far we have described the three development phases for our network model showing how criticality depends on network state, where the final state suggests some kind of fixed point behavior. In the following we will assess to what degree this process is characteristic for the system. To this end, we calculate its nullclines analytically

(

To be able to solve the problem analytically we assume that the change of the connectivity

When considering axons and dendrites separately, fixed point

Furthermore, as the rate essentially follows

The dynamic behavior shown in

(

Phase | parameter | initial | supercritical | subcritical | critical |

Model | exponent | ||||

Experiment | exponent | ||||

Thus, this predicts that that developing inhibition is an important factor for the course of criticality in developing neuronal networks. Only if inhibition in the model is lowered in Phase III again, the network becomes critical. Therefore, it is likely that overall synaptic pruning in developing networks not only affects excitatory but also inhibitory synapses

Additional inter-spike interval (ISI) and cross-correlation (CC) analyzes have been performed. ISIs and CCs are very similar between cultures and model across all stages but they do not contain interesting features (like oscillations) and therefore we do not show these diagrams here to save some space.

The following predictions are derived from the model:

Criticality at the end of development is optimally reached with 20% inhibition with a strength equal to that of the excitation. This observation does not depend crucially on the distribution of the inhibitory neurons and corresponds approximately to the normal degree of inhibition in cortical networks and cultured networks, respectively. A higher degree leads to sub- and a lower degree to supercritical behavior. A subcritical state is observed in cell cultures before they reach a mature state. This predicts that the onset of inhibition in the cultures must be strong with a connectivity much larger than in the end. Also the time-course of reaching firing rate homeostasis appears to be shorter than the curbing of this overly strong inhibition which takes longer and thus leads to the subcritical state observed in the interim. These are model predictions, because there is currently no data existing about the temporal development of inhibition. This data would be required to extend our model by implementing some time-course (dynamic coupling function) of the inhibitory development. Due to the lack of this data, this seems not useful at the moment because there is no way to constrain such a model extension. In general, the average homeostatic firing rate is independent of the level of inhibition and will in Phase III be reached regardless. All these predictions could be tested by measuring the degree of inhibition in the developing cultures and by pharmacologically interfering with the normal development forcing cultures to develop stronger (or weaker) inhibitory networks.

Average rate

For a network that has reached homeostasis, criticality can probably still be influenced by a sudden, prolonged change of inhibition. Following the second prediction, we expect some lasting connectivity changes to take place in these cases leading to a mildly changed criticality. Remarkably, first experimental evidence exists that an acute change in inhibition in fact alters criticality in mature cultured networks

The model further predicts that the actual balance between axonal supply

These predictions are quite specific as they do not depend on the parameter choices in the model, which is one strength of this approach. Most predictions, if not all, can be tested in a straight forward way in future experiments, albeit requiring substantial and sometimes difficult experimental work which can only be addressed in future work.

In the current study we have investigated how the activity patterns in developing cell cultures can be measured and modeled in terms of self-organized criticality. We have shown that the activity distributions in real cultures undergo a transition from a stage with little activity to a supercritical and then a subcritical state and finally to critical behavior. These transitions were significant for the cell cultures analyzed.

We used an extended version of the neurite outgrowth model by Van Ooyen and co-workers

In general the chosen abstractions in the model appear to match the data description level quite well, but the question arises to what degree this still corresponds to reality in developing networks. Most importantly, the network stages described above must be related to the morphological development of dissociated neurons and their growing connectivity in culture which determines the activity pattern at every point in time

It is well known

While certain simplifying assumptions had to be made to arrive at the basic differential equations (Equations 14–17) of our model, these experimental results clearly support the general dynamics assumed for our model.

In our model, networks with about 20% inhibition where the only ones that reached a robust critical state. While this level of inhibition corresponds to that in real nets, the results is intriguing as homeostasis of the firing rate will also be reached with much different levels of inhibitory cells. As known from literature

Like others

Self-organized criticality represents the situation that many systems of interconnected, nonlinear elements evolve over time into a critical state in which the probability distribution of avalanche sizes can be characterized by a power law. This process of evolution takes place without any external instructive signal. As analytically shown

The current study shows that networks in cell cultures undergo a certain transition during their morphological development. Thus, this paper is in the tradition of a sequence of investigations

Several previous studies

In a previous study Beggs and Plenz

Primary cortical cell cultures were prepared as described previously

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PKC |

Electrophysiological recordings were performed on the different DIVs at the same time for one hour under culture conditions with a MEA1060-BC system amplifier (Multi Channel Systems)

Clustering of neurons

A total of

order to assess the distribution of avalanches in cell cultures and in the model in the same manner, we search for the beginning and the end of an avalanche by a gliding time bin of a fixed size. Whenever the system is silent (no spikes) for at least the duration of the time bin, an avalanche ends and a new one starts with the next spike. The time bin is the mean time interval between two spikes in the system. Too long time intervals are sorted out by first calculating the mean cross-correlation of all electrode signals and secondly by getting the time value for which 99% of the integration area is under this mean cross-correlation curve. This time value is the maximum time interval which is taken into account of the mean time interval

To distinguish between the different states of an SOC system, the measure

These thresholds are heuristic as there is no theoretical background for this. In general they correspond well to state classification if made by human inspection. Note, the results of this paper are not crucial dependent on narrow threshold margins.

Measuring a power law for the avalanche distribution is not sufficient to conclusively show that a system is in the critical state, because a power law can also result from the summation of two exponentials

In (

A third informative test is to assess the scale-free behavior of the inter-avalanche intervals (

Finally, a fourth test for criticality is the Fano Factor

For the critical state in our model and cell cultures the Fano Factor shows a power law behavior for a wide range of time windows

In order to investigate the relationship between network development and self-organized criticality, we extended the previous neurite outgrowth model by Van Ooyen and Van Pelt

As in the main text in the following, mean values are given as upper case letters, while lower case letters indicate individual values. For a fixed connectivity, given by the synaptic density between dendritic and axonal offers, each neuron has a certain activity. In accordance to the definition of the neuron model in the work by Abbott and Rohrkemper

We model the calcium dynamics in our neuron model related to the work by Abbott and Rohrkemper

The development of dendrites and axons depends indirectly, by ways of the calcium concentration, on the activity. Lipton and Kater

The following table (

Parameter | |||||||||

Value | 0.0005 | 5 | 1000 | see text | 10 | 0.5 | 0.05 | 0.01 | 0.02 |

Derivation of the steady state firing rate R^{*}.

(0.03 MB PDF)

Derivation of the nullcline of the model.

(0.03 MB PDF)

We thank Anna Levina for helpful discussions about avalanche statistics.