Our Model of Structural Plasticity (MSP) (Butz and van Ooyen, 2013; Butz et al., 2014) represents synapses not merely as synaptic weight factors but as composed of two complementary synaptic elements: an axonal element representing axonal boutons or terminals, and a dendritic element representing any postsynaptic specialization on the dendrite (e.g., a dendritic spine). Synaptic elements develop independently of their matching element in an activity-dependent manner. A neuron creates new synaptic elements when its level of electrical activity is below a homeostatic set-point and decreases the number of elements when its activity exceeds this set-point. In addition, neurons need a minimum level of activity to form synaptic elements. Newly formed elements are vacant and available for synapse formation. Vacant axonal and dendritic elements can connect to form a new synapse. Synaptic elements of adjacent neurons are more likely to connect than those of more distant neurons. Vacant synaptic elements that are not used for synapse formation decay spontaneously with a certain rate. Existing synapses can break up if an element bound in a synapse is removed by the hosting neuron. The complementary synaptic element of the broken-up synapse becomes vacant and available for synapse formation again, which enables structural rewiring of neuronal networks. The algorithm proceeds in three steps. First, electrical activity is computed for every neuron. Second, numbers of synaptic elements are updated depending on the current average level of electrical activity of each neuron, which may cause the breaking of synapses. Third, vacant synaptic elements are recombined to form new synapses. Changes in electrical activity and number of synaptic elements proceed on a continuous timescale, whereas the breaking and formation of synapses take place at discrete time steps.

The same network and neuron model was used as in Butz and van Ooyen (2013), with n^ex = 320 excitatory and nⁱⁿ = 80 inhibitory Izhikevich neurons (Izhikevich, 2003). Inhibitory neurons only differ from excitatory ones in the sign of synaptic transmission. Excitatory neurons were placed with some jitter on a 20 x 16 grid with a spatial distance between two grid points of 150 μm. Inhibitory neurons were placed evenly between the excitatory neurons. Electrical activity is modeled by two differential equations, one for the membrane potential v and one for a recovery variable u enabling re-polarization after an action potential:

where v and u are in mV, t is in ms, k₁ = 0.04 mV⁻¹ms⁻¹, k₂ = 5 ms⁻¹, and k₃ = 140 mVms⁻¹. Every time a neuron fires (v ≥ 30 mV), v and u are reset:

where a = 0.1 ms⁻¹, b = 0.2 ms⁻¹, c = −65 mV, and d = 2 mVms⁻¹. Synaptic input I_syn has a fixed strength of 1 mVms⁻¹ for every synapse. Synaptic input arriving at the postsynaptic neuron is low-pass filtered by an exponential filter function h(t)=exp(−tμ) with decay constant μ = 5 ms. External input I_ext is permanently delivered as white noise with mean 5 mVms⁻¹ and standard deviation 1 mVms⁻¹ according to Izhikevich (2003); Butz and van Ooyen (2013).

Intracellular calcium concentration is used as a low-passed filtered average of the firing frequency of each neuron (Butz and van Ooyen, 2013). Every time a neuron fires, calcium concentration is increased by β = 0.001 ms⁻¹ and then decreases exponentially to zero with decay time τ_Ca = 10000 ms.

We used our model of structural plasticity (MSP), which is described in detail in Butz and van Ooyen (2013); Butz et al. (2014). The model proceeds in three steps: (1) updating electrical activity, as described above; (2) updating the number of synaptic elements and eventually the breaking of synapses if synaptic elements were deleted; and (3) the formation of new synapses.

We grew every model network from scratch, i.e., starting with zero connectivity and zero synaptic elements. Networks were formed by exactly the same growth rules that were effective after the lesion. However, in order to grow networks from scratch, it was necessary to use initially a higher level of external input. We used I_ext = 8 mVms⁻¹ for the first 500 updates in connectivity and then lowered it gradually down to 5 mVms⁻¹ according to I_ext(T) = ((8−5)/(1+exp((T−500)/200))+5) mVms⁻¹. At T = 8000, we removed the input of a circumscribed area, the lesion projection zone (LPZ), by setting I_ext,LPZ(T) = 0 (for T ≥ 8000) permanently. The LPZ spans from x1 = 5× 150 μm to x2 = 12× 150 μm and from y1 = 5× 150 μm to y2 = 12× 150 μm (cf. Figure 5) for all simulations and all cases (cf. Update of synaptic elements and breaking of synapses). We refer to the rest of the network with intact input as “intact zone.” Every simulation is continued for another T = 12000 updates in connectivity. As in our previous work (Butz and van Ooyen, 2013), we matched 1000 updates in connectivity with 14 days post-lesion. Thus, simulations predict the time course of network rewiring for 24 weeks after the lesion.

A neuronal network can be seen as a graph, with neurons as nodes and synapses as edges or links between nodes. Since the presynaptic neuron always activates the postsynaptic neuron (and never the other way round), we regard the graph as directed. In order to describe changes in network topology after a focal loss of input, we assessed the following graph theoretical measures at every update in connectivity. To reduce the complexity of the assessment, we considered only the topology of the excitatory synaptic connections W^ex,ex between the n^ex excitatory neurons. For the graph theoretical assessments, the brain connectivity toolbox by Rubinov and Sporns was used (Rubinov and Sporns, 2010).

In our previous work (Butz and van Ooyen, 2013), we postulated activity-dependent growth curves for axonal and dendritic elements that gave rise to the same kind of network rewiring as observed in primary visual cortex after focal retinal lesions. With these growth curves (referred to as physiological growth curves), in which axonal elements required higher levels of electrical activity than dendritic elements to grow out (η_A = 0.4, η_D = 0.1), the LPZ recovered from the outside to the inside and the turnover of dendritic elements was surprisingly similar to the experimental data on dendrtic spine turnover (Butz and van Ooyen, 2013). In the present study, we investigated how network topology changes in response to a focal loss of input, with neurons rewiring their inputs (and outputs) locally in order to restore a desired level of electrical activity. Our modeling results show that networks employing physiological growth curves return to a homeostatic range in electrical activity (Figure 2A) and, as a result of compensatory rewiring, become more randomly connected, as indicated by a lower value of the small-world parameter S (Figure 2B) measured over the entire network. Although random networks have no nodes of particular importance and hence a low betweenness centrality, neurons in the LPZ have a higher betweenness centrality after network rewiring than before the lesion (Figure 2C).

The decrease in small-world parameter S is determined by the course of the clustering coefficient γ and the characteristic path length λ. While λ converges to one, γ decreases markedly (Figure 3) and is thereby responsible for networks becoming more random. The decrease in clustering is not immediate but sets in between 6 and 8 weeks after the lesion. As will be shown below, it takes some time until network reorganization has managed to restore neuronal activities to their homeostatic range. During this time period, there is a temporary drop in characteristic path length below one, which contributes to a temporary rise in S (Figure 2B). However, after about 16 weeks, λ reaches stable values around one. From the same time on, S stabilizes at lower levels than in control networks without lesions.

From our previous work on modeling cortical rewiring after focal retinal lesions (Butz and van Ooyen, 2013), we know that functional network repair can be brought about by an, also experimentally observed, ingrowth of connections from the intact zone to the LPZ (Darian-Smith and Gilbert, 1994; Yamahachi et al., 2009). For physiological network repair to go along with functional retinotopic remapping (as shown in mice Keck et al., 2008), we found that it is important that the majority of new connections impinging on deafferentated neurons originates from intact areas and transmits electrical activity from the intact zone to the LPZ. Here, we further investigate whether the changes in global topology parameters express this ingrowth of connections. For this, we first focus on the activity-dependent changes in synapse numbers and connectivity between the intact zone and the LPZ. The first 6 weeks are dominated by a loss of synapses originating from the LPZ (Figure 4A). This is a direct consequence of neuronal activities being low and calcium concentrations being below η_A = 0.4 (Figure 4B), which causes axonal elements to be removed. By contrast, axonal elements from the intact zone form additional synapses with the LPZ right from the onset of the lesion. Between 6 and 8 weeks after the lesion, most neurons in the LPZ have reached calcium levels of 0.4 and start forming additional axonal elements, connecting to targets in the LPZ as well as the intact zone. The number of recurrent synapses from the LPZ to the LPZ does thereby at no time exceeds the number of synapses from the intact zone to the LPZ, as required for a functional remapping to emerge.

The change in λ and γ as shown in Figure 3 is measured over the entire network. We further want to understand whether the course of γ and λ is caused by the changing connectivity between the intact zone and the LPZ. For this, we assessed γ and λ for the set of LPZ and intact zone neurons separately. We can distinguish three phases in the time course of both parameters. These phases arise from the interaction between the loss of connections from the LPZ and the formation of new connections from the intact zone. The initial phase lasts for the first 4 weeks after the lesion and is dominated by a loss of connections from the LPZ. This is reflected by a decrease in γ, especially of LPZ neurons but to a lesser extent also of intact zone neurons (Figure 4C). At the same time, λ of paths from the LPZ to the intact zone increases (Figure 4D) due to the loss of connections from the LPZ to the intact zone. Conversely, λ of paths from the intact zone to the LPZ decreases because new connections are being formed originating from the intact zone.

During the second phase, roughly between 4 and 8 week, we see a temporal increase in γ of both the LPZ and the intact zone neurons (Figure 4C). This increase essentially contributes to the temporal increase in small-worldness of repairing networks as shown in Figure 2B. During this phase, the decrease in number of connections from the LPZ slows down, while new connections from the intact zone are still being formed. During this second phase, especially λ for paths from the LPZ to the intact zone shows a rapid decrease (Figure 4D). This rapid decrease is brought about by a few new connections that are formed as soon as LPZ neurons reach calcium levels of 0.4 (Figure 4B). This happens already slightly before the average number of synapses from the LPZ to the intact zone increases significantly at about 6 weeks after lesion.

A third phase can be distinguished from 8 weeks after the lesion onwards, when LPZ neurons start forming outgoing connections again. Especially the recurrent connections inside the LPZ (Figure 4A) lead to an increase in γ of LPZ neurons (Figure 4C), while neurons in the intact zone show a decrease in γ after the temporary rise. However, γ of the LPZ neurons is not strictly increasing over time; between 8 and 12 weeks after the lesion, γ decreases a second time before it finally increases toward a stable level. We can explain this fluctuation in γ by the ongoing replacement of connections during this period. Only if all neurons in the LPZ have reached calcium levels beyond 0.4, and hence contribute to axonal element and (outgoing) synapse formation, does the clustering coefficient strictly increase until rewiring comes to a standstill. During the third phase, λ of paths from intact zone to LPZ further decreases (Figure 4D). This further decrease is brought about by additional connections inside the LPZ, contributing to network repair and shortening paths to neurons in the LPZ. The decrease in path lengths to the LPZ also explains the increasing betweenness centrality of LPZ neurons, since betweenness centrality by definition is a measure of how many shortest paths go via certain nodes. As shown in Figure 4D, λ of paths from the LPZ to the intact zone takes on values of a randomized network.

Interestingly, the absolute clustering of neurons in the intact zone shows very little change (Figure 4E), implicating that the particular course of γ arises from changes in the number of connections and their clustering in comparison with a randomized network. By contrast, the changes in clustering of the LPZ (Figure 4E) as well as the characteristic path length for both the LPZ and the intact zone (Figure 4F) show similar courses for the non-normalized and normalized values. Therefore, we may conclude that networks become more random because of the increase in number of connections, whereas the increase in betweenness centrality (as a result of decreasing path lengths from the intact zone to the LPZ) is a consequence of added specific projections from the intact zone to the LPZ.

Network repair does not in all cases lead to the formation of synapses from the outside to the inside and a functional reorganization of connectivity. In our previous study (Butz and van Ooyen, 2013), we identified three different cases of network rewiring depending on the relative values of the growth parameters η_A and η_D. For η_A > η_D, we observed network repair in line with the exeperimental data (physiological case); for η_A = η_D = 0.1, we observed network repair brought about by massive recurrent connections (recurrent case); and for η_D > η_A, we observed no network repair at all (no-repair case).

The network rewiring occurring in the last two cases are referred to as aberrant network rewiring. Figure 5 depicts the most evident differences in the layout of connections after compensatory network rewiring between the physiological and the recurrent case and shows the no-repair case for the sake of completeness.

Whereas in the physiological case (Figure 5A) most of the newly formed synapses from the intact zone terminate in the LPZ, we do not see this ingrowing of new synapses in the recurrent case (Figure 5B) or in the no-repair case (Figure 5C). In the recurrent case and the no-repair case, new synapses from anywhere in the intact zone predominantly connect to neurons in the intact zone in the direct vicinity of the LPZ. In the pysiological and the recurrent case, but not in the no-repair case, LPZ neurons contribute to network repair by forming additional synapses. However, there is an important difference between the physiological and the recurrent case in where LPZ neurons project to. LPZ neurons in the physiological case form new connections to neurons in the LPZ and preferentially to those in its center (inset Figure 5A), whereas LPZ neurons in the recurrent case also project to neurons in the intact zone and show less projection preference (inset Figure 5B). A marked difference between the physiological and the recurrent case is seen in the loss of synapses originating from the LPZ. Whereas many synapses are lost in the physiological case, almost no synapses originating from the LPZ are eliminated in the recurrent case. Therefore, network repair in the recurrent case is brought about by addition of new synapses, whereas in the physiological case network repair goes along with a replacement of synapses. The no-repair case shows a considerable loss of synapses originating from the LPZ. Neurons in the LPZ are not able to raise their activity beyond η_A (Figure 6C) and therefore lose axonal elements and outgoing synapses as a direct consequence of the growth rules.

Whereas physiological network repair goes along with increasing randomness of network connectivity, as indicated by a decrease in small-world parameter S, we do not see a considerable change in S in the recurrent case (Figure 6A). Interestingly, in networks that lack repair after lesions, we even see an increase in S. The strongest increase in betweenness centrality of neurons in the LPZ is observed in the recurrent case (Figure 6B) and sets in much earlier (after about 2 weeks) than in the physiological case due to a mere addition of synapses rather than a replacement of synapses. Betweenness centrality goes to zero (Figure 6B) when neurons do not return to the homeostatic range in activity (Figure 6C). Given that in the recurrent case, neurons in the LPZ restore their activity most quickly and completely and with the strongest increase in betweenness centrality, we may conclude that the increase in betweenness centrality is an indicator for the success of network repair in terms of restoring neuronal activity.

Local and global efficiency are additional measures quantifying changes in network topology (Equations 13 and 14). Global efficiency indicates how efficiently information can travel through the entire network; i.e., global efficiency is the averaged sum of the inverse of the shortest paths between any two neurons in the entire network. By contrast, local efficiency of neuron i measures how efficiently information can be exchanged among neurons that are connected to neuron i; i.e., local efficiency is the averaged sum of the inverse of the shortest path between any two neurons connected to neuron i (excluding neuron i itself). Especially in sparsely connected networks, efficiency as a topology measure is preferred over characteristic path length and clustering coefficient because it can be meaningfully computed also for unconnected neurons. In the physiological case, we observe a decrease in local efficiency (Figure 7A) but an increase in global efficiency (Figure 7C) relative to the efficiencies before the lesion. Both local and global efficiency go through an initial phase in which they decrease, reaching a minimum at about 6 weeks after the lesion. The global efficiency recovers and finally even exceeds its initial level, whereas the local efficiency recovers little and remains lower than before the lesion. By contrast, in the recurrent case, both local and global efficiency increase immediately after the lesion and exceed by far their initial levels and the levels in the physiological case. A drop in local and global efficiency is observed when no network repair takes place. The intact zone does not show a considerable change in either local or global efficiency (Figures 7B,D). The ratio of local to global efficiency indicates the relative amount of local clustered and global long-range connectivity. The stronger increase in global than in local efficiency in the physiological case reflects the increasing ramdomness (cf. Figure 2B), whereas recurrent networks with a strong increase in global and local efficiency become even more small-world (cf. Figure 6A).

The stronger increase in local efficiency in the recurrent case compared with the physiological case is brought about by the massive formation of partly recurrent connections originating from the LPZ. In fact, the number of recurrent synapses in the LPZ exceeds by far the number of synapses from the intact zone to the LPZ and from the LPZ to the intact zone (Figure 8A). The high number of recurrent synapses leads to a strong increase in clustering coefficient (Figure 8B). The clustering coefficient of the LPZ after rewiring even exceeds that of the intact zone; the latter does not change notably after the lesion. Remarkably, the average shortest paths from the intact zone to the LPZ and those from the LPZ to the intact zone strongly decrease simultaneously (Figure 8C).

The different types of network rewiring have a direct impact not only on global network topology but also on the local degree distributions of neurons. Before the lesion, neurons of the intact zone and the LPZ have the same in- and out-degree distribution, in the physiological case (Figures 9A,B) as well as in the recurrent case (Figures 10A,B). The distributions in the physiological case are slightly more tailed than in the recurrent case. After the lesion in the physiological case, the center of the in-degree distribution of the LPZ neurons shifts to the right (Figure 9C), indicating the presence of neurons with high in-degrees. The centers of the out-degree distributions of LPZ and intact zone neurons do not change, but the distributions as a whole become more fat-tailed (Figure 9D). Thus, compensatory network rewiring generates more hub-like neurons in the LPZ, but the majority of neurons in the LPZ and the intact zone maintains its out-degree. In the recurrent case, LPZ neurons shift the centers of their in- and out-degree distributions completely to the right (Figure 10), but the distributions do not become more fat-tailed. The in- and out-degree distributions become markedly different from the ones before the lesion and from the degree distributions of the intact zone neurons. We may conclude that due to the massive recurrent connections, the LPZ neurons separate from the intact zone in terms of degree distribution.

Network repair is not dependent on a particular initial network topology. The networks considered so far have a high clustering and a low characteristic path length before the lesion (small-world networks). However, even random networks with low initial clustering and characteristic path length fully restore their average electrical activity (in terms of calcium concentration) back to the homeostatic range, regardless of whether growth rules of the physiological case (Figure 11A) or the recurrent case (Figure 11D) are used. In random networks, the activity of the LPZ does not decrease so strongly as in clustered networks, because vacant axonal elements are available from anywhere in the network and network repair is immediately effective. The fastest restoration of electrical activity is seen for the recurrent case with random networks (Figure 11D). In addition to the availability of axonal elements from anywhere in the network, in the recurrent case neurons with low activity also provide their own vacant axonal elements, contributing to fast network repair.

Irrespective of growth rules and initial network topology, restoration of firing rates is accompanied by an increase in betweenness centrality (Figures 11B,E). For the physiological and the recurrent case, betweenness centrality reaches higher absolute values in small-world networks than in random networks. However, the greatest increase in betweenness centrality is seen for the recurrent case with random networks. Interestingly, for all scenarios studied (Figures 11A,D), the strongest increase in betweenness centrality is associated with the fastest restoration of electrical activity. We conclude that the increase in betweenness centrality is a generic effect of compensatory network rewiring because it is independent of initial connectivity and strongly correlates with effectiveness of network repair, in terms of speed and completeness of restoring electrical activity. Moreover, the physiological case with small-world networks is the only scenario in which topology becomes more random (Figure 11C). In all other scenarios (Figures 11C,F), we see only little change and initially random networks become only slightly more structured (small increase in S) after the lesion.

Remarkably, network reorganization in the model shows striking similarities with intracortical network reorganization on a mesoscopic scale [e.g., a retinotopic remapping with filling of the LPZ from the outside to the inside (Butz and van Ooyen, 2013); mesoscopic defined as in Liljenstroem, 2001] and may also account for macroscopic network changes after, for example, focal, subcortical stroke. An impairment of motor function of the hand after subcortical stroke coincides with a loss in effective connectivity of inter-area cortical motor networks, especially between pre-motor and primary motor cortices in the hemisphere ipsilateral to the stroke site (Grefkes et al., 2008). Conversely, restoration of electrical activity and functional recovery are associated with increasing effective connectivity from prefrontal to motor cortices (Sharma et al., 2009). The functional effects are thought to arise from a loss of local connections within the motor network and the formation of additional long-range excitatory connections from prefrontal to motor areas (Sharma et al., 2009). In our model, we also observe the local removal of connections from the deafferentated neurons in the LPZ and an ingrowth of more long-range connections from the intact zone into the LPZ. Note that in the model as well as in the brain, the loss of local connections is not a mere consequence of degeneration caused by the primary lesion but a secondary effect due to network reorganization (Rehme et al., 2011).

In stroke patients, even brain regions remote from the lesion site change their topology (Carmichael, 2006). However, not the entire brain changes its topology but only those regions directly connected to the primary lesion site (Carmichael et al., 2001; Dancause et al., 2005; Rehme and Grefkes, 2013). Provided that connected brain regions become deafferentated by the primary lesion site, homeostatic structural plasticity, as revealed by our modeling study, may account for the observed changes in macroscopic topology after a lesion, i.e., an increased randomness of network connectivity and an increased or decreased betweenness centrality of particular regions (Wang et al., 2010; Shi et al., 2013). Wang et al. (2010) reported an increase in betweenness centrality of brain regions that became deafferentated by a subcortical stroke, namely the ipsilesional primary motor area and the contralesional cerebellum. Note that the predominant cortico-ponto-cerebellar fiber tract crosses in the brainstem to the contralateral side, and a subcortical lesion will therefore deafferentiate the contralesional cerebellum. By contrast, the contralesional primary motor cortex and the ipsilesional cerebellum decreased their betweenness centrality. The latter two regions are not directly affected by deafferentation but are still involved in reorganization, since especially contralateral areas seem to support their homotopic regions by compensatory sprouting during stroke recovery (Carter et al., 2010). An increase in betweenness centrality of deafferentated brain regions and a decrease in betweenness centrality of brain regions supporting recovery perfectly match with our model findings, so we hypothesize that the observed topological changes in the brain of stroke patients may be accounted for by homeostatic structural plasticity. Furthermore, in the model, the increase in betweenness centrality has proven to be the strongest indicator of network repair under different conditions. Therefore, increasing betweenness centrality could be a biomarker for brain repair after lesions such as stroke. In future work, we intend to implement homeostatic structural plasticity in a large-scale model of micro- and macroscopic connectivity containing multiple brain regions (Potjans and Diesmann, 2014).

The time course of changes in topology in our self-repairing network model shows remarkable similarities with the time course of topology changes during brain repair, especially in patients with subcortical stroke. Subcortical stroke involves, apart from damage to a circumscribed volume of brain tissue, a loss of input to other brain regions, particularly those of the motor network. The model predicts a pronounced increase in small-worldness of the entire network during the initial phase of compensatory network rewiring, before the network in the end becomes more random. Indeed, brain networks after subcortical stroke increase their small-world property in the subacute phase (about 1 week post-infarct) (van Meer et al., 2012) and thereafter become continuously more random. As in our model, the change in small-worldness of brain networks is brought about by a marked change in clustering.

From our model we further predict that right after the lesion, local as well as global efficiency drops markedly as a result of loss of connections. The decrease in efficiency is in agreement with changes in network topology observed after stroke (Honey and Sporns, 2008; Alstott et al., 2009). In the model, local efficiency remains always lower than before the lesion, but global efficiency increases markedly and reaches values higher than before the lesion. Strikingly, even in well-recovered stroke patients, brain networks are found with low local but high global connectivity (Rehme and Grefkes, 2013). However, brain network topology with low local and high global efficiency may contribute to less stable performance of sensorymotor skills (Rehme and Grefkes, 2013).

Interestingly, in the model we observe only small changes in topology within the first 4 weeks. Network repair in stroke patients is also not immediate. From monkey studies it is well known that it takes about 7–14 days after stroke until axonal sprouting occurs, and new connections are visible not before 28 days (Carmichael, 2003), with lesion-induced network rewiring continuing for at least 3–6 months (Carmichael, 2006; Cramer, 2008). The time course of axonal sprouting in the experiment is comparable to the time course of axonal element formation in our model and also matches the physiological time course of structural plasticity in mice (Keck et al., 2008; Butz and van Ooyen, 2013). The model illustrates that network repair after deafferentation and stroke can be brought about by local, homeostatic growth rules.

The time course of network repair is determined by the relation between the growth curve parameters η_A and η_D. If axonal elements require more activity to form than dendritic elements (i.e., η_A > η_D), networks will show a compensatory growth of connections from the intact zone into the LPZ. However, if axonal and dendritic elements grow at the same low level of activity, deafferentated neurons will literally pull them selves by their own bootstraps by forming massive recurrent connections to restore activity to the homeostatic set-point (Butz and van Ooyen, 2013). By contrast, the course of reorganization is not crucially dependent on the particular choice of parameters for neuronal electrical activity (Figure S1), as long as the network is able to reach a homeostatic equilibrium before the external input is removed. Other parameters, such as the width of the kernel σ = 150 μm, have been chosen in agreement with experimental findings (De Paola et al., 2006). Likewise, the decay time of intracellular calcium was chosen to be of the same order of magnitude as measured experimentally (Hofer et al., 2011).

The notion that neurons strive to restore their level of electrical activity after loss of input is now widely accepted (Hengen et al., 2013; Keck et al., 2013). Even in stroke, the need of neurons to restore electrical activity to a homeostatic set-point may be an underlying principle of recovery (Avramescu et al., 2009). Today, the predominantly discussed mechanism for maintaining homeostasis in electrical activity is synaptic scaling (Turrigiano and Nelson, 1998). However, synaptic scaling restores activity within 48 h, yet network reorganization continues massively for several months. Therefore, synaptic scaling cannot be the only mechanism involved in network reorganization after deafferentation and stroke. Moreover, Hengen et al. (2013) showed that firing rates in V1 after focal retinal lesions restore within the first 48 h but drop again thereafter before they slowly rise again. This finding is in line with previous reports on the extended time course of network repair (up to 12 months) after focal retinal lesions and the restoration of electrical activity from the outside to the inside of the LPZ (Giannikopoulos and Eysel, 2006). In the first 48 h after the lesion, homeostatic synaptic scaling may upregulate firing rates (Keck et al., 2013), but the continuing structural changes in connectivity beyond 48 h may alter activity levels and may bring neurons again outside their homeostatic range of activity. Rewiring connectivity may provide a straightforward explanation for the experimental observation that activity levels drop again after 48 h (Hengen et al., 2013) and slowly recover over several weeks (Giannikopoulos and Eysel, 2006; Hu et al., 2009, 2010). In the model, the removal of connections is accounted for by the minimal levels of activity needed for maintainance and formation of axonal and dendritic synaptic elements (η_A and η_D, respectively); required minimal levels of activity have not been discussed in recent concepts of homeostatic plasticity (but see van Ooyen et al., 1996). In our model as well as in visual cortex after focal retinal lesions, activity slowly recovers over periods of weeks and months (Giannikopoulos and Eysel, 2006; Butz and van Ooyen, 2013). We hypothesize that the increase in firing rates is due to the ingrowth of long-range connections from intact regions, which may be guided by homeostatic structural plasticity. Therefore, we postulate that homeostatic plasticity is more than just synaptic scaling and needs to be extended to encompass structural plasticity, including the reorganization of synaptic connections. With the present model, we have provided growth rules that can govern homeostatic structural plasticity and that can lead to physiologically realistic network reorganization on a microscopic, mesoscopic (Butz and van Ooyen, 2013), and macroscopic level.

Partial deafferentation, as caused by focal stroke for example, can lead to epileptiform activity and seizures (Topolnik et al., 2003; Avramescu and Timofeev, 2008). It has been discussed that homeostatic synaptic plasticity may contribute to post-traumatic epileptogenesis in chronically isolated cortex (Houweling et al., 2005). Synaptic scaling (Turrigiano and Nelson, 1998), a well-studied mechanism for homeostatic synaptic plasticity, is known to generate epileptiform activity (Froehlich et al., 2008). However, synaptic plasticity does not include the rewiring of networks and acts on timescales of hours rather than weeks or months. Although a previous modeling study (Houweling et al., 2005) has suggested that anatomical network rewiring is not required for epileptiform activity to occur, we argue that without network rewiring an important aspect of lesion-induced plasticity is left out. For example, models without structural plasticity cannot account for the clinical observation that although spontaneous seizures are most frequent within months after the lesion, they can occur up to 5 years post-lesion (Temkin, 2001). Therefore, we propose that synaptic scaling may account for spontaneous seizures early after the lesion but that for the pathogenesis of post-traumatic epilepsy months after the lesion, homeostatic structural plasticity may be a more suitable explanation (see also van Oss and van Ooyen, 1997).

In the model, a change in the value of just a single parameter, namely the level of activity needed for axonal elements to form (η_A), leads to massive recurrent connections, which, as we showed in a previous study (Butz and van Ooyen, 2013), can generate strongly synchronized activity patterns comparable to epileptiform activity. In an in vitro injury model of epilepsy, Srinivas et al. (2007) showed that epileptogenesis goes along with a marked increase in connectivity [also supported by findings on recurrent mossy fiber sprouting in an organotypic cell culture model of hippocampal epilepsy (Kharatishvili et al., 2007)] and that the shape of the degree distribution of the neurons changes from power-law to Gaussian. Interestingly, in the recurrent case of our model, which generates epileptiform activity after network reorganization, the degree distribution of LPZ neurons is much less tailed than in the physiological case after network repair. Therefore, we hypothesize that the way brain networks rewire after lesions determines whether or not patients develop post-traumatic epilepsy. This notion is further supported by the finding that the shape of the lesion can affect epileptogenesis (Volman et al., 2011), since it is more likely that the shape of the lesion can influence epileptogenesis with growth of new connections than with synaptic scaling. Importantly, our model predicts that the sensitivity of axonal outgrowth to low levels of activity might be decisive for whether recurrent connections with epileptiform activity, or physiological network repair with normal activity patterns, emerge after brain lesions. This insight may help find novel molecular targets for pharmacological treatments to prevent post-traumatic epilepsy, which are urgently needed as post-traumatic epilepsy is often impervious to medical treatment (Herman, 2002; van Breemen et al., 2007).

Homeostatic structural plasticity is a new concept for network reorganization, with large implications for understanding and stimulating brain repair after lesions. Models of homeostatic structural plasticity can help integrate recent clinical findings on changing brain topology after a variety of pathologies, including stroke, Alzheimer's disease and multiple sclerosis. These models can assist us in uncovering the mechanisms underlying functional reorganization and in finding biomarkers for successful brain repair, such as an increased betweenness centrality of brain regions deprived of input from primary lesion sites. Most importantly, however, homeostatic structural plasticity puts functional reorganization of brain networks into a different light. The predominant dogma of plasticity is still Hebbian plasticity, with its “fire together, wire together” slogan (Hebb, 1949). With Hebbian plasticity, enforcing (synchronous) activity strengthens synapses. By contrast, the homeostatic nature of structural plasticity implies the need for a moderate level of activity, because the formation of axonal and dendritic structures is maximal for activity levels slightly below a desired set-point of electrical activity. We postulate that the brain has the highest plasticity for recovery when neurons and brain regions, especially those supporting deafferentated regions in the recovery process, have not yet returned to their homeostatic equilibrium. We might call this initial phase a critical period for brain repair, in analogy to critical periods in neural development. During network development, too, neurons shape their connectivity until desired activity levels are reached (Tetzlaff et al., 2010). As a consequence, in neurorehabilitation treatment, not only stimulation by physical training or direct electrical stimulation but also pauses in treatment may be important. Stimulation may increase electrical activity beyond the homeostatic set-point, inducing pruning of existing synaptic connections, whereas treatment pauses may lower activity and bring activity levels into an optimal range for the formation of new connections (Butz et al., 2009a). Moreover, network reorganization does not always need to be functional; as our model suggests, post-traumatic epilepsy could be the result of miss-wiring or over-compensation. Treatments must therefore focus more on the time course and current state of network repair. Lastly, large-scale computer models, such as those developed in the context of the human brain project (www.humanbrainproject.eu) will, once structural plasticity has been incorporated, be valuable tools in finding and testing treatment strategies for patients with brain damage.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.