Growth Rules for the Repair of Asynchronous Irregular Neuronal Networks after Peripheral Lesions

Several homeostatic mechanisms enable the brain to maintain desired levels of neuronal activity. One of these, homeostatic structural plasticity, has been reported to restore activity in networks disrupted by peripheral lesions by altering their neuronal connectivity. While multiple lesion experiments have studied the changes in neurite morphology that underlie modifications of synapses in these networks, the underlying mechanisms that drive these changes are yet to be explained. Evidence suggests that neuronal activity modulates neurite morphology and may stimulate neurites to selective sprout or retract to restore network activity levels. We developed a new spiking network model, simulations of which accurately reproduce network rewiring after peripheral lesions as reported in experiments, to study these activity dependent growth regimes of neurites. To ensure that our simulations closely resemble the behaviour of networks in the brain, we deafferent a biologically realistic network model that exhibits low frequency Asynchronous Irregular (AI) activity as observed in cerebral cortex. Our simulation results indicate that the re-establishment of activity in neurons both within and outside the deprived region, the Lesion Projection Zone (LPZ), requires opposite activity dependent growth rules for excitatory and inhibitory post-synaptic elements. Analysis of these growth regimes indicates that they also contribute to the maintenance of activity levels in individual neurons. Furthermore, in our model, the directional formation of synapses that is observed in experiments requires that pre-synaptic excitatory and inhibitory elements also follow opposite growth rules. Lastly, we observe that our proposed model of homeostatic structural plasticity and the inhibitory synaptic plasticity mechanism that also balances our AI network are both necessary for successful rewiring of the network.

)) are initially connected via synapses with a connection probability of (p = 0.02). All synapses (EE, EI, II), other than IE synapses, which are modulated by inhibitory spike-timing dependent plasticity, are static with conductances g EE , g EI , g II , respectively. All synapse sets are modifiable by the structural plasticity mechanism. External Poisson spike stimuli are provided to all excitatory and inhibitory neurons via static synapses with conductances g E ext and g I Inh , respectively. To simulate deafferentation, the subset of these synapses that project onto neurons in the Lesion Projection Zone (LPZ) ( (Figure 1a). Apart from inhibitory synapses projecting from the inhibitory As in Butz and van Ooyen's MSP framework, each neuron possesses sets of both pre-synaptic (axonal) and post-synaptic (dendritic) synaptic elements, the total numbers of which are represented by (z pre ) and (z post ), respectively. Excitatory and inhibitory neurons only possess excitatory (z E pre ) and inhibitory axonal elements (z I pre ), respectively, but they can each host both excitatory and inhibitory dendritic elements (z E post , z I post ) (Figure 2a). The rate of change of each type of synaptic element, (dz/dt), is modelled as a Gaussian function of the neuron's "calcium concentration" ([Ca 2+ ]): Here, ν is a scaling factor and η, define the width and location of the Gaussian curve on the x-axis.
Extending the original MSP framework, we add a new parameter ω that controls the location of the curve on the y-axis. The relationship between η, and the optimal activity level of a neuron, ψ, govern the activity-dependent dynamics of each type of synaptic element. A neuron should not turn over neurites when its activity is optimal ([Ca 2+ ] = ψ). This implies that the growth curves must be placed such that dz/dt = 0 when [Ca 2+ ] = ψ. Hence, ψ can take one of two values: (ψ = η) or (ψ = ), and the turnover of synaptic elements dz/dt is: This is illustrated in Figure 2. Other than in a window between η and where new neurites sprout, 116 they retract. The new parameter, ω, permits us to adjust the speed of sprouting and retraction 117 (Figures 2b and 2c). In Figure 2b with (ψ = η), new neurites will only be formed when the neuron 118 experiences activity that is greater than its homeostatic value (ψ < [Ca 2+ ] < ). Figure 2c, on the 119 other hand, shows the case for (ψ = ), where growth occurs when neuronal activity is less than 120 optimal (η < [Ca 2+ ] < ψ).

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The [Ca 2+ ] for each neuron represents a time averaged measure of its electrical activity: V th is the threshold membrane potential. shows mean firing rate of neurons in LPZ-C; (c) shows mean firing rate of neurons in peri-LPZ; (d) shows the coefficient of variation (CV) of the inter-spike intervals of neurons in the LPZ-C and peri-LPZ. The graph is discontinuous because ISI CV is undefined in the absence of spikes in the LPZ C; (e) shows spike times of neurons in the LPZ C and peri-LPZ over a 1 s period at t = {1500 s, 2001.5 s, 4000 s, and 18 000 s}. The network is permitted to achieve its balanced Asynchronous Irregular (AI) low frequency firing regime under the action of inhibitory synaptic plasticity (t 1500 s). Our structural plasticity mechanism is then activated to confirm that the network remains in its balanced AI state (panel 1 in Figure 3a). At (t = 2000 s), neurons in the LPZ are deafferented (panel 2 in Figures 3a and 3e are at t = 2001.5 s) and the network allowed to repair itself under the action of our structural plasticity mechanism (panels 3 (t = 4000 s) and 4 (t = 18 000 s) in Figures 3a and 3e). The balance between excitation and inhibition (E-I balance) received by a neuron may be disturbed by a change in either of the two types of input. Post-synaptic elements of a neuron react to deviations in activity from the optimal level (ψ) by countering the changes in excitatory or inhibitory inputs to restore the E-I balance. For both excitatory and inhibitory neurons, excitatory post-synaptic elements sprout when the neuron experiences a reduction in its activity, and retract when the neuron has received extra activity. Inhibitory post-synaptic elements for all neurons follow the opposite rule: they sprout when the neuron has extra activity and retract when the neuron is deprived of activity. (b) pre-synaptic elements. In excitatory neurons, axonal sprouting is stimulated by extra activity. In inhibitory neurons, on the other hand, deprivation in activity stimulates axonal sprouting. Synaptic elements that do not find corresponding partners to form synapses (free synaptic elements) decay exponentially with time. These graphs are for illustration only. Please refer to Table 2 for parameter values.    Figure 3b). Experimental studies report that these neurons gain excitatory synapses on newly formed dendritic spines [Kec+08] and lose inhibitory shaft synapses [Che+12] to restore activity after deprivation. The increase in lateral excitatory projections to these neurons requires them to gain excitatory dendritic (postsynaptic) elements to serve as contact points for excitatory axonal collaterals. At the same time, inhibitory synapses can be lost by the retraction of inhibitory dendritic elements. This suggests that new excitatory post-synaptic elements should be formed and inhibitory ones removed when neuronal activity is less than its optimal level (([Ca 2+ ] < ψ) in Figure 4a): While we were unable to find experimental evidence on the activity of excitatory or inhibitory neurons just outside the LPZ, in our simulations, these neurons exhibit increased activity after deafferentation (t = 2000 s in Figure 3c). Unlike neurons in the LPZ that suffer a net loss of excitation, these neurons appear to suffer a net loss of inhibition, which indicates that they must gain inhibitory and lose excitatory inputs to return to their balanced state. Hence, the formation of new inhibitory dendritic elements and the removal of their excitatory counterparts occurs in a regime where neuronal activity exceeds the required amount (([Ca 2+ ] > ψ) in Figure 4a): The constraints described by equations 2, 4, and 5 can be satisfied by Gaussian growth rules for ex-154 citatory and inhibitory dendritic elements, with E post = ψ and η I post = ψ, respectively ( Figure 4a).

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Given the distinct characteristics of excitation and inhibition, the two growth rules were treated in-156 dependently and the parameters governing them were tuned iteratively over multiple simulation 157 runs. For example, sufficiently high values for the rate of formation of inhibitory dendritic elements 158 had to be selected for excitatory neurons to prevent the build up of excessive excitation (Table 2).
159 Figure 5 shows the time course of rewiring of excitatory and inhibitory connections to excita-160 tory neurons in the centre of the LPZ that results from the growth curves in our simulations. As  in excitatory or inhibitory input received by a neuron also relies on the availability of pre-synaptic 176 counterparts. We derive activity dependent growth rules for excitatory (z E pre) and inhibitory (z I pre ) 177 pre-synaptic elements in a similar manner to that used for post-synaptic elements.

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Within the LPZ, the increase in excitation requires a corresponding increase in the supply of excitatory pre-synaptic elements. Experimental evidence reports a sizeable increase in the formation and removal of axonal structures in and around the LPZ [Yam+09], with a marked addition of lateral projections from neurons outside the LPZ into it [Mar+10]. While an increase in pre-synaptic elements within the LPZ may contribute to repair, an inflow of activity from the periphery of the LPZ to its centre has been observed in experiments [DG94; Kec+08; Mar+10], pointing to the inwards sprouting of excitatory axonal projections from outside the LPZ as the major driver of homeostatic rewiring. For this sprouting of excitatory projections from the non-deafferentated area into the LPZ to take place in our simulations, the increase in activity in neurons outside the LPZ must stimulate the formation of their excitatory axonal elements: Conversely, neurons outside the LPZ with increased activity need access to inhibitory pre-synaptic elements in order to receive the required additional inhibitory input. Deafferentation studies in mouse somatosensory cortex [Mar+10] report more than a 2.5 fold increase in the lengths of inhibitory axons projecting out from inhibitory neurons in the LPZ two days after the peripheral lesion. This outgrowth of inhibitory projections preceded and was faster than the ingrowth of their ]. In our simulations, the experimentally observed outward protrusion of inhibitory axons from the LPZ requires that the formation of inhibitory pre-synaptic elements is driven by reduced neuronal activity: Similar to the post-synaptic growth rules, the pre-synaptic growth rules for excitatory and in-179 hibitory neurons were also treated separately and their parameters were tuned iteratively over re-  (Table 2). A neuron in its steady state receives excitatory (g E ) and inhibitory (g I ) conductance inputs through its excitatory (z E post ) and inhibitory (z I post ) dendritic elements, respectively, such that its activity ([Ca 2+ ]) is maintained at its optimal level (ψ) by its net input conductance (g net ). (b) An external sinusoidal current stimulus (I ext ) is applied to the neuron to vary its activity from the optimal level. (c) Under the action of our post-synaptic growth curves, the neuron modifies its dendritic elements to change its excitatory (∆g E ) and inhibitory (∆g I ) conductance inputs such that the net change in its input conductance (∆g net ) counteracts the change in its activity: an increase in [Ca 2+ ] due to the external stimulus is followed by a decreas in net input conductance through the post-synaptic elements and vice versa (dashed lines in Figures 8b and 8c). received by the neuron (g net ), which modulates its activity, can be estimated as the difference of the 202 total excitatory (g E ) and inhibitory (g I ) input conductances.

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The activity of the neuron is then varied by an external sinusoidal current stimulus (Figure 8b). In

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Simulations where homeostatic synaptic plasticity was disabled, on the other hand, also failed 233 to re-establish the balanced state of the network before the peripheral lesion ( Figure 9c, and 9d).

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While the activity of the deprived neurons in the LPZ initially increased back to pre-lesion levels, 235 under the action of structural plasticity only, the network eventually started exhibiting abnormally 236 high firing rates instead of settling in the desired low firing rate regime. These results suggest that 237 inhibitory synaptic plasticity is required to finely tune inputs to neurons so that the network can 238 achieve its balanced state.

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Thus, our simulations predict that both homeostatic processes are required for successful repair- anced asynchronous irregular network (Figure 1 and 2). We show that our simulations reproduce

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On the pre-synaptic side, axonal turnover and guidance has been investigated in much detail, 276 and is known to be a highly complex process incorporating multiple biochemical pathways [LV09; 277 Goo13]. Our hypothesis regarding excitatory pre-synaptic structures is supported by a report by and structural plasticity, our simulations also require both structural and synaptic plasticity for suc-307 cessful network repair (Figure 9). Thus, our simulation results lend further support to the notion 308 that multiple plasticity mechanisms function in a cooperative manner in the brain.

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As a computational modelling study, our work necessarily suffers from various limitations. For  in GitHub repositories here and here (these repositories are currently private). The scripts used to 362 analyse the data generated by the simulation are available in a separate GitHub repository here.

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These repositories are licensed under the Gnu GPL license (version 3 or later). The data generated 364 by the simulations has been made available here (the data will be uploaded to a service suggested 365 by the reviewers, such as Zenodo).
where C is the membrane capacitance, V is the membrane potential, g L is the leak conductance, g exc 370 is the excitatory conductance, g inh is the inhibitory conductance, E L is the leak reversal potential, 371 E exc is the excitatory reversal potential, E inh is the inhibitory reversal potential, and I e is an external 372 input current. Incoming spikes induce a post-synaptic change of conductance that is modelled by 373 an exponential waveform following the equation: where τ g is the decay time constant andḡ is the maximum conductance as the result of a spike at 375 time t s . Table 1 enumerates the constants related to the neuron model.
Here, ν is a scaling factor, ξ and ζ define the width and location of the Gaussian curve on the x-axis, while ω controls the location of the curve on the y-axis (0 < ν, 0 < η < , 0 < ω < 2). Given that ([Ca 2+ ] > 0), (dz/dt) is bound as: Within these bounds, as shown in Figure 2, (dz/dt) is: If, based on its activity, a neuron has more synaptic elements of a particular type (z) than are cur-384 rently engaged in synapses (z connected ), the free elements (z free ) can participate in the formation III: synaptic plasticity only Figure 10. The simulation runs in 2 phases. Initially, the setup phase (0 s < t < t 2 ) is run to set the network up to the balanced AI state. At (t = t 2 ), a subset of the neuronal population is deafferented to simulate a peripheral lesion and the network is allowed to organise under the action of homeostatic mechanisms until the end of the simulation at (t = t end ). Each homeostatic mechanism can be enabled in a subset of neurons to analyse its effects on the network after deafferentation.
of new synapses at the next connectivity update step: However, if they remain unconnected, they decay at each integration time step with a constant rate 387 τ free : On the other hand, a neuron will lose z loss synaptic connections if the number of a synaptic element 389 type calculated by the growth rules (z) is less than the number of connected synaptic elements of 390 the same type (z connected ):    The simulation is then started and the network permitted to stabilise to its balanced AI state until 419 (t = t 2 in Figure 10). This phase consists of two simulation regimes. Initially, only inhibitory 420 synaptic plasticity is activated to stabilise the network (t < t 1 in Figure 10).   Figure 1b).

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• LPZ B: the inner border of the LPZ (Yellow in Figure 1b).

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• Other neurons: neurons further away from the LPZ (Grey in Figure 1b).

Network reorganisation 439
The deafferented network is permitted to reorganise itself under the action of the active homeostatic 440 mechanisms until the end of the simulation (t = t end in Figure 10). By selectively activating the two 441 homeostatic mechanisms in different simulation runs, we were also able to investigate their effects 442 on the network in isolation. which are considered immune to activity dependent changes in stability. They are removed from the 486 list of options from which z loss synapses are selected for deletion and are therefore, not considered 487 for deletion at all.

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For simplicity, for static excitatory synapses that all have similar conductances (EI, EE), we do not 489 use this method of deletion. Instead, for these, z loss connections are randomly selected for deletion 490 from the set of available candidates. While II synapses are also static, the deletion of an inhibitory 491 synapse by the loss of an inhibitory post-synaptic element can occur by the removal of either an IE 492 or an II synapse. Therefore, to permit competition between II and IE synapses for removal, we apply 493 weight based deletion to both these synapse sets.

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The numbers of synaptic elements are updated at every simulator integration time step inter-495 nally in NEST. Connectivity updates to the network, however, require updates to internal NEST 496 data structures and can only be made when the simulation is paused. This increases the computa-497 tional cost of the simulation, and we only make these updates at 1 s intervals. Gathering data on 498 conductances, connectivity, and neuronal variables like [Ca 2+ ] also require explicit NEST function 499 calls while the simulation is paused. Therefore, we also limit dumping the required data to files to 500 regular intervals. Table 4 summarises the various synaptic parameters used in the simulation.  502 We also studied the effects of our structural plasticity hypotheses in individual neurons using single 503 neuron simulations. Figure 8a shows a schematic of our single neuron simulations.

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The neuron is initialised to a steady state where it exhibits an indegree similar to neurons in 505 the network simulations when in their AI state. To do so, a constant baseline input current I ext 506 is supplied to the neuron to provide it with activity. The [Ca 2+ ] obtained by the neuron at this 507 time is assumed as its optimal level, ψ. Using identical values of η and but different ν values for 508 excitatory and inhibitory post-synaptic elements (ν E post = 4ν I post to mimic the initial indegree of 509 neurons in our network simulations), and an input current that deviates the activity of the neuron 510 off its optimal level (< I ext ), the neuron is made to sprout z E post , z I post excitatory and inhibitory 511 post-synaptic elements respectively (z E post = 4z I post ). By assuming that each dendritic element 512 receives inputs via conductances as observed in network simulations (g EE , g IE ), the net input to the 513 neuron that results in its activity can be approximated as: At this stage, the neuron resembles a one in network simulations in its balanced state before 515 deafferentation. The current input is returned to its baseline value, thus returning the [Ca 2+ ] to 516 its optimal value, ψ. In addition, the growth curves for the neuron are restored as per our activity 517 dependent structural plasticity hypotheses to verify that the neuron does not undergo any structural 518 changes at its optimal activity level.

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The external current input to the neuron is modulated sinusoidally to fluctuate the neurons 520 [Ca 2+ ] (Figure 8b), and resultant changes in the numbers of its post-synaptic elements are recorded.

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As the neuron modifies its neurites, the change in excitatory and inhibitory input conductance re-522 ceived as a result is calculated (Figure 8c).