Network model

We place N = m × m neurons on a regular square lattice of size L with spacing h = L/(m − 1). The spatial coordinates of a given neuron i that lies at lattice index (i_x, i_y) are (1)

and are small random jitters in the x- and y- coordinates of the neurons. The neurons are enumerated as i = i_x + (i_y − 1)m, where i = 1, …, N.

The membrane potential of i-the neuron, is governed by [21] (2) where C is the membrane capacitance, g_leak,i is the leak conductance, and V_rest is the resting membrane potential. The time-varying synaptic conductance, g_syn,i(t), and the reversal potential, V_syn, determine the time-varying synaptic inputs from the presynaptic partners. I_stim,i(t, l_i,r) is the stimulation current received by the neuron i at a distance l_i,r from the r-th stimulation electrode [detailed in the ’Stimulation’ section], which could be set to 0 for all i to study the stead-state dynamics of the network. Lastly, I_noise,i represents the noisy inputs from the other sources, e.g., other neuronal populations not included in the model. A neuron fires a spike whenever its membrane potential crosses the dynamic threshold, V_th,i, governed by (3) τ_th is the threshold time constant and V_th,rest is the resting threshold potential. Once the neuron generates a spike, its membrane potential, V_i(t), is kept at V_spike and V_th,i(t) is kept at V_th,spike for a duration of τ_spike. After that, the membrane potential is reset to V_reset.

The synaptic conductance, g_syn,i, follows (4)

Here, τ_syn is the synaptic time constant, κ is the maximal coupling strength, t_j,μ is the timing of the μ-th spike of the j-th presynaptic neuron, and t_d is the synaptic time delay. In general, multiple contacts can exist from neuron j to neuron i, the weights of which are given by the elements, w_i,j,k(t), of the weight matrix, W(t), where is the maximum number of contacts permitted between any pair of pre- and post-synaptic partners, and index k refers to the specific contact from neuron j and neuron i. The sum over k in Eq 4 refers to the summation over A_i,j contacts, where A_i,j(t) is the element of the N × N adjacency matrix, A(t). The adjacency matrix determines the network structure such that A_i,j(t) assumes a value >0 if synaptic contacts from neuron j to i exist, and 0 otherwise. Therefore, index i refers to a postsynaptic neuron and j to its presynaptic partner. No self-loops are allowed, i.e., A_i,i(t) = 0. The time-dependencies of the weight and adjacency matrices correspond to the weight and structural plasticity, respectively.

The noisy input from other sources is modeled as independent Poisson spike train with the constant rate f_noise, and is given by (5) where the synaptic noise conductance, g_noise,i(t), follows (6)

In Eq 6, the second term represents Poisson noise with the noise intensity κ_noise. The summation is over Poisson spikes, where t_i,μ is the timing of the μ-th Poisson spike fed to neuron i.

STDP

We use the additive STDP rule as in Ref. [21]. The change in the weight of the synaptic contacts from a presynaptic neuron j to its postsynaptic partner i is given by (7)

Thus, w_i,j,k(t) → w_i,j,k + δw_i,j(q) ∀k. Here, q = t_i − t_j − t_d is the time lag between the spikes of the neurons i and j. η ≪ 1 ensures a longer timescale of synaptic weight change compared to neuronal membrane potential dynamics. τ₊ is the long-term potentiation (LTP) time constant and τ_R scales the long-term depression (LTD) time constant relative to τ₊. The total synaptic weight change due to all possible values of time lag, . Thus, b determines the asymmetry of the STDP rule, such that b < 1 makes STDP potentiation dominant, b > 1 makes it depression dominant, and b = 1 makes it balanced. The parameters of the STDP rule, including the time delay t_d are kept the same as in Refs. [21, 79], given in Table 1. Eq 7 is implemented as a set of differential equations for the weight matrix, W(t), and the traces χ(t), ψ(t) for pre- and post-synaptic spike trains [80], (8) where t_i,μ is the timing of the μ-th spike of the postsynaptic neuron and t_j,μ is that of its presynaptic partner. t_i and t_j represent the latest spike times that trigger potentiation and depression, respectively.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Parameters of the network model and stimulus.

https://doi.org/10.1371/journal.pcbi.1012261.t001

SP

Computational models of SP may include all aspects, from neurite outgrowth and retraction [81] to the generation and deletion of synaptic elements to the probabilistic formation of synaptic contacts [61]. BvOSP [49, 61] is a model that generates and deletes synaptic elements based on the average activity level of neurons and makes them available for synapse formation without modeling the activity-dependent outgrowth or retraction of neurites in order to form contacts and allows for multiple contacts between a pair of pre- and post-synaptic neurons, i.e., the adjacency matrix elements, A_i,j ∈ {0, 1, 2, ‥}. Its algorithm is presented in detail in the S1 Text and Refs. [49, 61, 82]. It has been used to generate networks from completely unconnected neurons spontaneously, reproduce experimentally observed network reorganization after lesion [49, 82], and explain clinically observed therapeutic effects of CR [20], transcranial direct current stimulation [83], and transcranial magnetic stimulation [84]. In a recent study, BvOSP was used to show that hSP is responsible for biphasic changes in the connectivity of pyramidal neurons in mouse anterior cingulate cortex 24 and 48 hours after optogenetic stimulation [85].

The inclusion of BvOSP in computational studies aimed at investigating the evolution of networks with both weight and structural plasticity over long periods of time may require considerably long computation time, which could limit the extent to which the impact of SP and variations in its control parameters on the network dynamics and properties can be studied. Establishing a simpler model of SP that captures the essential mechanisms that govern changes in brain networks may allow for a reliable and more detailed numerical analysis. We propose a simple model, stochastic SP (described below), which directly builds or eliminates synaptic contacts between neurons without separately modeling the neurite outgrowth/retraction or generation/deletion of synaptic elements. We further simplify the model by allowing only a single synaptic contact from a given presynaptic neuron to its postsynaptic partner that accounts for its overall effect on the postsynaptic one, i.e., A_i,j ∈ {0, 1}, and , although it can be easily extended to include multiple synaptic contacts between a pair of pre- and post-synaptic neurons. We validate the stochastic SP method by comparing our results with those obtained using BvOSP, where we allow for multiple contacts with so that A_i,j ∈ {0, 10}.

We base our stochastic SP model on the experimental evidence stated above. The probability of the addition of a synaptic contact between two neurons, P_add, depends on the Euclidean distance between them and the firing rate of the postsynaptic neuron. Primarily, the postsynaptic neurons with firing rates below the homeostatic set-point (target) firing rate develop synaptic contacts with nearby neurons. The synaptic contacts are pruned depending on both synaptic weight and the firing rate of postsynaptic neurons, which together determine the pruning probability, P_prn. In particular, the synaptic contacts that are weaker or deliver inputs to neurons with a firing rate above the target firing rate are more likely to be pruned, consistent with experimental observations [41, 42]. The addition of a synaptic contact from a neuron j to i changes the adjacency matrix element A_i,j from 0 to 1 while pruning changes A_i,j from 1 to 0. The newly added contacts are given a small random weight, following experimental evidence [41].

The dependence of the pruning and addition probabilities on the postsynaptic neuron’s firing rate models a homeostatic process whereby the neuron’s activity is maintained at a target firing rate, f_T. The probability of addition of incoming contacts increases with a decrease in the neuron’s firing rate below f_T, while its pruning probability increases with an increase in the neuron’s firing rate above f_T. The firing rate of the i-th neuron is calculated by low-pass filtering its spike train as, (9) where t_i,μ is the timing of the μ-th spike of i-th neuron and τ_slow is the time constant.

The process of addition is purely homeostatic, i.e., weight-independent. The probability of addition is given by (10) where P_h is the maximal probability of the homeostatic SP (both addition and pruning), l₀ is the decay constant for distance dependence, and is the Euclidean distance between neurons i and j. Since the neurons are arranged on L × L square lattice, the network structure becomes distance-independent for . The firing rate dependence function, G(.), is a logistic function defined as (11) where the parameters Ω₀ and ν determine the midpoint and the slope, respectively.

In our model, a synaptic contact can be pruned independently by either the homeostatic or weight-dependent processes. The homeostatic component of the pruning is given by (12)

We assume non-zero probabilities of addition and pruning when the firing rate of a post-synaptic neuron matches its target rate, f_T, i.e., . The steepness of the logistic function is characterized by the parameter Δf, such that , and , for pruning and addition, respectively. This gives

The weight-dependent component of pruning is of the form (13) where w_min ≪ 1 and P_w is the maximal probability of weight-dependent pruning. The total probability of pruning is, (14)

Fig 1 exemplifies the SP probabilities. Since only one contact can be built from neuron j to i, in our stochastic SP model, the subscript k in Eqs 13 and 14 can be dropped.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. SP probabilities for the typical parameter values.

A: Probabilities of homeostatic SP versus the firing rate of a post-synaptic neuron. The vertical dashed line marks the target firing rate, f_T. B: Total SP probabilities versus synaptic weight according to Eqs 10 and 14 when the firing rate of the neuron equals the target rate. The parameters are: Δf = 1 Hz, , f_T = 4.5 Hz, w_min = 0.001, l = 0.1, l₀ = 0.5; P_h = 1 for A and P_h = 0.01, P_w = 1 for B.

https://doi.org/10.1371/journal.pcbi.1012261.g001

Measures

The degree of synchrony of a network is measured using the Kuramoto order parameter [21, 86], (15) where ϕ_i(t′) is the phase of i-th neuron that is calculated using the timings of two consecutive spikes, t_i,μ and t_{i,μ + 1},

ranges from 0 for complete absence of synchrony to 1 for perfect synchrony.

We calculate the network-averaged firing rate, 〈f〉(t) = (1/N)∑_i f_i(t) and the coefficient of variation, CV(t), of firing rate defined as the ratio of the standard deviation of firing rate, , to the mean, 〈f〉(t): CV(t) = σ_f(t)/〈f〉(t). The combination of order parameter and CV of the firing rate can be used as indicators of synchrony and frequency locking for non-identical oscillators. CV = 0, combined with the order parameter close to 1 indicates a perfectly frequency-locked state. Non-zero CV values indicate non-identical frequencies, e.g., due to incomplete synchronization and/or cluster states with different frequencies.

The synaptic weights are studied using the average incoming synaptic weight, W_i(t), of individual neurons and the network-averaged synaptic weight, 〈W〉(t) (16)

〈W〉(t) ∈ [0, 1], where 0 indicates an uncoupled network and 1 a strongly coupled one. 〈W〉(t) = 0 (1) may indicate desynchrony (synchrony). W_i → 1 (0) if neuron i is strongly (weakly) driven by its presynaptic partners via its incoming contacts. We also study the distribution of the synaptic weight of individual contacts in the steady synchronized states of the network.

To analyze the structure of the network, we use incoming and outgoing node degree densities (in-NDD and out-NDD) of individual neurons, given by (17) and the network-averaged NDD, . β(t) can range from 0 for a fully unconnected network to 1, marking an all-to-all connected network. For ease of use, we drop the superscript for the in-NDD hereafter, i.e., .

We employ the Pearson correlation coefficient to characterize the assortativity of the networks [56, 87, 88]. In (dis)assortative networks, the nodes (here, neurons) tend to connect to other nodes with (dis)similar properties on average. We determine the (dis)assortativity of a network for in- and out-degrees of the neurons as follows [89]. Let ϵ, υ ∈ {in, out} be the degree-type and and be the corresponding degrees of the pre- and post-synaptic neurons, respectively, connected by the e-th edge (contact). The Pearson correlation coefficient is given by (18) where E = βN(N − 1) is the total number of contacts. The averages, and . The standard deviations, and . We calculate the Pearson correlation coefficient between the in-NDDs of the pre- and post- synaptic neurons (in-in), the out-NDDs (out-out), and between the in-NDD of presynaptic neurons and the out-NDD of postsynaptic neurons (in-out) and vice versa (out-in) [89–91].

Negligible time variations in the network-averaged measures characterize the steady states. We use the network-averaged firing rate, 〈f〉, and synaptic weight, 〈W〉, to determine the approach to a steady state as follows. We integrate the model equations in time intervals of T = 60 s until both 〈W〉 and 〈f〉 converge with a given relative accuracy, Υ, (19) where n = 1, 2, …, n_min, ‥, n_max is the number of the convergence intervals (iterations). n_min is the minimum number of intervals and n_max is the maximum number of iterations required to achieve the relative accuracy of Υ = 10⁻³.

Stimulation

Stimulation is used to induce therapeutic effects in pathological conditions, such as epilepsy [92–94] and Parkinson’s disease [72, 75, 95, 96], where a synchronized state is associated with pathology while incoherent spiking of neurons is observed in a healthy state.

We use a multichannel stimulation protocol, UMRS (illustrated in Fig 2), where N_s electrodes at fixed locations deliver stimulation independently. Each stimulation site, r, receives uncorrelated stimulus at random times with exponentially distributed inter-pulse intervals, similar to the temporal randomness of the RR stimulation [21, 22]. The stimulus from the r-th electrode received by neuron i at a distance l_i,r from the electrode is given by a_sI₀X_r(t)D(l_i,r), where X_r(t) is the charge-balanced stimulation current of the r-th electrode. The dimensionless parameter a_s ∈ [0, 1] scales the magnitude, I₀, of the stimulus, and D(l_i,r) determines the spatial drop in stimulus with distance from the electrode. The total amount of stimulus received by i-th neuron is (20)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. UMRS stimulus administered at N_s = 4 sites.

The sites of stimulation are chosen to be the centers of the 4 quadrants in the square network plane. The left panel shows the example stimulus pattern, X(t), with F_s = 100 Hz for a 50 ms duration. Each site of stimulation receives an independent stimulus and the time interval between stimulus events (each event comprising a positive pulse followed by a negative pulse after a short gap) is exponentially distributed. The right panel shows the spatial variation of stimulus with distance from the electrodes, controlled by D(l_i,r), for all 4 electrodes placed at the centers of four quadrants. Neurons are marked with crosses at their positions.

https://doi.org/10.1371/journal.pcbi.1012261.g002

The time-dependent charge-balanced stimulus current, X_r(t), is a random sequence of stimulus events consisting of positive and negative pulses. Each rectangular positive pulse of duration and amplitude 1, followed by a short gap, , and then by a rectangular negative pulse of duration and amplitude . The value of the magnitude I₀ is chosen as . Intervals between excitatory pulses are exponentially distributed with the mean inter-pulse interval τ_UMRS. We imposed a minimum interval between subsequent stimulus events at τ_Λ = 7.692 ms, corresponding to a maximum stimulation frequency of 130 Hz [21, 22]. Thus, the mean stimulation frequency, F_s = (τ_UMRS + τ_Λ)⁻¹.

For scaling the stimulus with distance, we use a Gaussian function, , for simplicity. The parameter σ_s determines the area of stimulus spread and can be associated with the number of neurons that effectively receive the stimulus. For a fraction, γ, of neurons or the area of the network that we intend to stimulate with each electrode, the value of σ_s can be determined as follows: Assume that below 1% of the maximal intensity, the stimulus is not effective, i.e., it does not affect the spiking of neurons. The area covered by the stimulus from each electrode is π(3σ_s)² ≈ γL², where L is the network size, since D(l_i,r) drops to ≈ 0.01 at l_i,r = 3σ_s. Therefore, (21) since .

Parameters and implementation

The parameters of LIF neurons were the same as in [21]. Table 1 contains the values of the model parameters, unless stated otherwise. To generate the Gaussian distributed firing rates of unconnected neurons with the network-averaged firing rate f₀ and SD σ_f, the leak conductance of neurons, g_leak,i, was drawn from a Gaussian distribution with mean and standard deviation . The parameter σ_f quantifies the diversity of neurons and was varied in this study. and were obtained numerically for a given noise intensity, κ_noise, as follows. First, we calculated and tabulated the long-time average firing rate of a single neuron as a function of g_leak, f(g_leak), for values of g_leak in the range 0.005 ≤ g_leak ≤ 0.05 mS/cm². Second, we best linear fitted f(g_leak), as f(g_leak) = α₁g_leak + α₂. The regression coefficients, α₁ and α₂, were then used to calculate

Now, if f is a random Gaussian variable with the mean f₀ and SD σ_f, then the equation above can be considered as a linear transformation of f. Then, g_leak is also Gaussian random variable with the mean and SD given by

For noise intensity, κ_noise = 0.06 mS/cm², used in this study, α₁ = 125.67 and α₂ = 0.92.

In this study, we separately consider networks with

1. only synaptic weight plasticity, labeled as STDP-only;
2. homeostatic SP (hSP) alone, i.e., without weight-dependent pruning (P_w = 0) and without STDP, labeled as hSP-only;
3. a combination of STDP and hSP, i.e., without weight-dependent pruning (P_w = 0), labeled as STDP+hSP;
4. a combination of STDP and SP that includes hSP and weight-dependent pruning (P_w ≠ 0), labeled as STDP+SP.

For networks with STDP-only, the neurons are connected on a random graph with a probability equal to the desired average NDD, i.e., p = β₀ = β(0), on a square lattice of size L = 1 mm. The probability for two neurons to connect decreases exponentially with distance as exp(−l/l₀). The initial membrane potentials of the neurons are drawn from a uniform distribution between V_reset and V_rest. The threshold potential, V_th,i is set to V_th,rest mV, and g_syn,i and g_noise,i are set to 0 for all neurons. The convergence to steady state is determined using Eq 19, where n_min is set to 60, and n_max can be as large as required for the convergence.

SP operates on a timescale much longer than those of STDP and neuronal spiking dynamics. The computation time required to implement structural updates at every time step or in short intervals of a few milliseconds can be significantly long and limit the detailed study of networks [20, 61]. It is crucial to make computation time shorter for thorough variations of parameters to investigate the network dynamics. Therefore, we separate the structural updates from the spiking dynamics and the synaptic weight change by considering structural updates as discrete events, similar to a previously used strategy [53]. Specifically, we assume that between consecutive structural updates (SP iterations) no structural change occurs for a time interval ΔT_sp and the network evolves either with STDP-only or no plasticity. Further, we require the time interval ΔT_sp to be long enough to let the network settle in a steady state according to the conditions of Eq 19 with n_min = 2 and n_max = 31, so that where . Thus, the SP time arrow is measured in units of ΔT_sp, T_sp,n′ = n′ΔT_sp, where n′ is the number of SP iterations.

In the following, we consider 20 × 20 networks of N = 400 neurons. Larger networks with N = 1024 neurons (32 × 32) yielded qualitatively similar results; an example of time evolution of network measures for N = 1024 is presented in Fig I in S1 Text. All simulations for spontaneous network dynamics were repeated and averaged for 10 realizations of random networks.

Networks with STDP-only

We study the steady-state dynamics that emerge spontaneously in a random network of LIF neurons with STDP-only for various initial conditions determined by the initial values of the network-averaged synaptic weight, 〈W〉(0), and NDD, β₀. The steady-state measures for the given initial conditions averaged over ten realizations of the random network of identical neurons (σ_f = 0) are shown in Fig 3 (top panels).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Steady states of the networks of identical (σ_f = 0 Hz, top) and non-identical neurons (σ_f = 0.5 Hz, bottom) with STDP-only.

The colors indicate the network-averaged values of the firing rate in A and D, the order parameter in B and E, and the CV of the firing rate of neurons in C and F for various values of the initial average synaptic weight, 〈W〉(0) and NDD, β₀. The results are averaged over ten realizations of the random networks for each combination of 〈W〉(0) and β₀. Other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g003

When the network is sparse, β₀ ≤ 0.05, it settles in a desynchronized state regardless of 〈W〉(0). For β₀ ≈ 0.05 and above, the network portrays multistability by settling in either desynchronized, partially synchronized, or fully synchronized states as indicated by the order parameter in Fig 3B. Interestingly, while intermediate values of 〈W〉(0) and β₀ produce strongly synchronized states with a minimal CV of firing rates and a high order parameter, larger 〈W〉(0) and β₀ cause the network to be only partially synchronized as implied by the higher CV of firing rates in Fig 3C and a lower order parameter in Fig 3B. The raster plot for a synchronized state observed for β₀ = 0.07 in Fig 4 shows minor spike-time jitters while that for a partially synchronized state observed at β₀ = 0.2 it shows significant spike-time jitters, illustrating reduced synchrony with increased β₀ and suggesting the optimal values of NDD for maximum synchrony.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Spike-raster plots of the network of identical neurons with STDP-only.

The left panel shows the raster plot for average NDD, β = 0.07, and the right panel shows the same for β = 0.2. For both cases, the network started with an initial average weight of 0.8. Other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g004

The fully synchronized states have a higher network-averaged firing rate than the desynchronized states and even higher firing rates emerge for partially synchronized states, accompanied by higher CV of firing rates. This is caused by the clustering of some neurons at higher firing rates than the network average. Fig 5C shows that the neurons in the denser network (β₀ = 0.2) form 4–5 clusters of the time-averaged firing rate. The firing rates of the clusters depend on their average incoming synaptic weight; naturally, the neurons with larger average incoming weights have higher firing rates. The map of time-averaged firing rates of individual neurons reveals the formation of high-frequency islands for β₀ = 0.2 [Fig 5B] that represent a chimera-like state. The fully synchronized state for β₀ = 0.07 is characterized by a single firing rate ≈ 4.2 Hz, as shown in Fig 5A and 5C.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Cluster formation in the network of identical neurons with STDP-only.

A and B: Time-averaged firing rates versus spatial location of neurons for the two indicated values of average NDD, β₀. C: Time-averaged firing rates versus average synaptic weight. For both cases, the network started with an initial average weight of 0.8. Other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g005

Next, we study the network of non-identical neurons with STDP-only. Neurons are made non-identical in an unconnected state by introducing diversity in the natural firing rates of the neurons, controlled by the diversity parameter, σ_f. Similar to the network of identical neurons above, the network of non-identical neurons exhibits multistability as illustrated in Fig 3(bottom panels) where the maps reveal that the ranges of 〈W〉(0) and β₀ for which the network settles in a desynchronized state extend to higher values than those for the network of identical neurons due to the diversity in natural firing rates. Nevertheless, the network in synchronized states has a higher network-averaged firing rate than the desynchronized states similar to the network of identical neurons, and the maximum synchrony is observed for intermediate values of 〈W〉(0) and β₀ where the firing rate CV is minimum. An increase in 〈W〉(0) and β₀ produces partially synchronized states with higher network-averaged firing rate and CV of the firing rate. This is further illustrated in Fig 6A, 6B and 6C. The network-averaged firing rate increases with an increase in the NDD, while the order parameter and the CV of firing rates show a non-monotonous dependence on NDD. Below a critical value of the average NDD, the order parameter resides at a low value, indicating a desynchronized state. Past the critical average NDD, the value of the order parameter increases and reaches a maximum value. A further increase in NDD causes the order parameter to drop to lower values, indicating partial synchrony. The CV of firing rates remains high for small and large values of average NDD and drops to a minimum at intermediate values when the network is strongly synchronized. At small average NDDs, the neurons fail to influence each other and thus, the network does not get synchronized. At larger average NDDs, groups of neurons with larger coupling fire at higher rates than the others, forming clusters and resulting in a high CV of firing rates. The corresponding parametric plots in Fig 6D and 6E show that the maximally synchronized state is obtained at a specific network-averaged firing rate where the maximum order parameter and the minimum firing rate CV are observed coincidentally, indicating maximum phase- and frequency-locking. For example, for σ_f = 0.5 Hz, the network is maximally synchronized at ≈ 4.5 Hz firing rate. A higher firing rate that occurs at a larger average NDD, results in a less coherent state with a higher value of firing rate CV. For a lower σ_f, the network gets synchronized with a lower average NDD and firing rate and tends to have a lower CV of firing rates than a network with a higher σ_f. The range of average NDD that is large enough to enable synchronization but not enough to promote clustering of neurons is smaller for networks of non-identical neurons compared to those of identical neurons since the identical neurons can get synchronized at significantly lower values of average NDD with a low average firing rate, while clustering occurs at large average NDDs, making the plateaus in Fig 6B, 6C, 6D and 6E larger for identical neurons.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. The synchronized state can be optimized by tuning the average NDD for the network with STDP-only.

A–C: Network-averaged firing rate, order parameter, and firing rate CV versus average NDD, β₀, for the indicated values of diversity parameter, σ_f. D and E: Order parameter and firing rate CV versus network-averaged firing rate, obtained as parametric plots of panels A–C. For each value of β the network was initialized with the average weight 〈W〉₀ = 0.8 and the results were averaged over 10 realizations of random networks; the error bars show the corresponding standard deviation. Other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g006

Effect of plasticity mechanisms on transition to synchronized states

Networks with hSP-only.

To develop and study the evolution of the network with hSP-only, we begin with completely unconnected neurons, i.e., β₀ = 0, and set a target firing rate, f_T. hSP begins to add synaptic contacts with random weights between 0 and 1 in order to bring the firing rate of all the neurons to the target level. The network-averaged firing rate, thus, increases as the average number of contacts, represented by the network-averaged NDD, β(t), increases over SP iterations and both saturate together to the steady-state values as shown in Fig 7A and 7C. Naturally, a higher average NDD is needed and therefore attained for a higher f_T. A low average NDD may prevent the network from reaching a synchronized state and a high average NDD may allow it to get synchronized. Thus, for smaller f_T the network ends up in a desynchronized state, e.g., for f_T = 3 Hz. For a sufficiently large f_T, the network may reach synchrony while for some intermediate f_T it may reside in a partially synchronized state, e.g., f_T = 4.5 Hz (or 6 Hz) and 4 Hz, respectively, as shown in Fig 7B. The convergence of firing rates of all neurons in the network to a higher f_T requires more SP iterations, as indicated by the delayed fall of CV for higher f_T in Fig 7D. During the transition to the target rate, the neurons have different firing rates, preventing them from getting synchronized. Therefore, the network requires more SP iterations for F_T = 6 Hz than for F_T = 4.5 Hz to get synchronized in Fig 7B. The synchronized states are accompanied by a significantly reduced CV of firing rates as shown in Fig 7D. The network-averaged firing rate overshoots the target when f_T = 3 Hz because the natural firing rate of the neurons is Gaussian distributed with mean 3 Hz and standard deviation 0.5 Hz. The neurons with natural firing rates below 3 Hz, develop new incoming contacts in order to reach the target firing rate while the ones above 3 Hz do not have any pre-existing contacts to lose, raising the network-averaged NDD and firing rate. The CV of firing rates is determined by the distribution of natural firing rates and the target rate. When the target rate is higher than the natural firing rate of all the neurons (f_T = 4.5 Hz and 6 Hz in this case), hSP enables all the neurons to attain the target rate and thereby reduces the CV to a minimum. A higher order parameter for f_T = 6 Hz compared to 4.5 Hz can be attributed to a higher network-averaged NDD. For smaller f_T, the neurons with natural firing rates above f_T remain unconnected, effectively reducing the order parameter and increasing the CV simultaneously. We observed a similar evolution of network-averaged measures using BvOSP in place of stochastic SP [Fig A in S1 Text].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Time evolution of the state measures as the network develops and reaches steady states with hSP-only.

The network-averaged firing rate (A), the order parameter (B), the network-averaged NDD (C), and the firing rate CV (D). The solid lines show the values averaged over 10 realizations and the shaded area represents the error bars. The diversity parameter is σ_f = 0.5 Hz and other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g007

Fig 8 shows that in a steady state for a given f_T, the number of presynaptic partners of neurons, represented by in-NDD, decreases with an increase in the natural firing rate below f_T, while the ones above f_T may not develop incoming contacts. hSP does not directly control the number of postsynaptic partners of neurons and thus, out-NDD displays no dependence on the natural firing rate. The resulting out-NDD is binomially distributed, due to which the spread of out-NDD is larger for larger average NDD, obtained for higher f_T. With BvOSP, we observe a similar dependence of in-NDD on the natural firing rate of the neurons as with our stochastic SP. However, since BvOSP determines both in- and out-NDDs identically based on the natural firing rate of the neurons, the out-NDD shows a similar dependence on the natural firing rate as the in-NDD, shown in Fig B in the S1 Text, unlike that with the stochastic SP.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Dependence of in- and out-NDDs on the natural firing rate of the neurons for the network developed with hSP-only.

The panels A-C correspond to the target firing rates, f_T, specified at the top. All the values are averaged over 10 realizations of random networks. The diversity parameter is σ_f = 0.5 Hz and other parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g008

Networks with combinations of structural and weight plasticity.

Here we develop networks using STDP+hSP and STDP+SP (with P_w = 0.01 and P_w = 1) and compare their transition to synchronized states with the network developed using hSP-only. With STDP+hSP and STDP+SP, the new contacts added at each SP iteration (structural update) are given a random initial weight ∈ [0, 0.2] and are subjected to STDP allowing those to get potentiated or depressed. Thus, the neurons can adjust their firing rates by adding and losing contacts besides changing the synaptic weights. The target firing rate is set to 4.5 Hz. This f_T corresponds to the network-averaged firing rate of a maximally synchronized random network with STDP-only, for which the order parameter and the firing rate CV attain their respective maximum and minimum [green curves in Fig 6D and 6E]. Since the target firing rate is higher than the natural firing rates of most of the neurons (mean 3 Hz and standard deviation 0.5 Hz), the initial structural updates involve only the addition of new contacts. The network may get synchronized when a sufficiently large number of contacts have developed with progressing SP iterations. This is indicated by an abrupt increase of the order parameter in Fig 9A and a drop of the firing rate CV in Fig 9C. With hSP-only, the network may require a larger number of contacts to get synchronized because some of the strong contacts that are added may not support but rather oppose synchronization. However, for networks with STDP+hSP and STDP+SP, the synaptic weight change of the contacts may preclude the need to build more contacts, allowing for synchronization at a significantly earlier stage of network evolution and, thus, for smaller average NDD compared to the network with hSP-only. Combining Fig 9A and 9B and by excluding the SP iteration axis provides the dependencies of the order parameter on β in Fig 9D. Similarly, the dependencies of the firing rate CV on β in Fig 9E are obtained by combining Fig 9C and 9B. We add the corresponding curves of the order parameter and the firing rate CV versus average NDD for random networks with STDP-only generated for every value of average NDD, as for Fig 6, to compare the transition of networks developed from scratch to random networks with STDP-only as well. Fig 9D clearly shows that networks with STDP+SP and STDP+hSP transition to their synchronized states for significantly lower average NDD (β ≈ 0.05) than networks with STDP-only (β ≈ 0.075) or hSP-only (β ≈ 0.1). The structural changes continue after the transition to synchronization, and the average NDD converges to a steady state value. In this post-synchronous stage of network evolution, the weight-dependent pruning becomes important. As we demonstrate in the next subsection, the synaptic contacts that become potentiated and promote synchronization tend to survive while the others that become weak get eliminated, thereby optimizing the network structure. Fig 9B shows that as the networks pass the onset of synchronization, the increase in average NDD remains smaller for the networks with STDP+SP compared to those with STDP+hSP and hSP-only and the steady state value of the average NDD decreases with the increase of the probability of weight-dependent pruning, P_w. This is further demonstrated in Fig 9D and 9E, where the curves of order parameter and CV versus β cease to change at smaller values of β for networks with STDP+SP. Importantly, Fig 9E shows that the plasticity of network structure results in synchronized states with much smaller values of the firing rate CV, indicating stronger frequency locking.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Comparison of network evolution in the presence of different plasticity cases.

The change in the order parameter, network-averaged NDD, and firing rate CV with SP iterations (A, B, and C, respectively). Order parameter and the CV of the firing rate versus the network-averaged NDD for five plasticity cases (D and E). The asymmetry parameter, b, of STDP, is set to 1, so that the desynchronized steady state is unstable [21]. The solid lines show the values averaged over 10 realizations and the shaded area represents the error bars. Parameters: P_h = 0.01, σ_f = 0.5 Hz.

https://doi.org/10.1371/journal.pcbi.1012261.g009

In summary, the transition to synchronized states in the presence of any combination of structural and weight plasticity with any form of SP may occur with significantly sparser networks compared to random networks with STDP-only or those generated using hSP-only. Additionally, hSP and SP that includes hSP and weight-dependent pruning reduce the firing rate mismatch of the neurons, characterized by smaller values of the firing rate CV.

Effect of including SP on steady synchronized random networks with STDP

Enhancement of synchronization.

In our model, synchronized random networks with STDP-only tend to have a higher CV of firing rates than the networks that evolve with any plasticity scheme that includes any form of SP [see Fig 9E]. We investigate the effect of taking such synchronized random networks with STDP-only through SP iterations, as described in ‘Parameters and implementation’ section. In the presence of SP, neurons tend to become identical as their firing rates converge to the given target rate by adding or losing synaptic contacts, reflected in the CV of firing rates. This may enable the neurons to get more strongly synchronized. We consider synchronized random networks with diversity parameter, σ_f = 0.5 Hz, and set the target firing rate f_T = 4.5 Hz as above. Fig 10 shows the evolution of the network-averaged measures for three values of initial average NDD, β₀ = 0.075, 0.1, and 0.2, as the networks progress through structural updates with STDP+SP (P_w = 0.01). With SP iterations, the firing rate evolves towards the f_T, a higher degree of synchrony as indicated by larger values of the order parameter, and a smaller value of the firing rate CV in Fig 10A, 10B and 10D, respectively, for all initial average NDDs. The average NDDs of the networks continue to adjust until the target rate is achieved, as shown in Fig 10C. Since f_T and the probabilities of addition and pruning are the same for all three initial average NDDs, the networks settle with the same average NDD in the steady state, although the underlying network structure may differ. We obtain similar results using BvOSP in place of stochastic SP even though it allows for multiple synaptic contacts between each pair of pre- and post-synaptic partners; the results are presented in Fig C in S1 Text.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Evolution of network-averaged measures with SP iterations.

Evolution of the initially random synchronized networks that progress through SP iterations using STDP+SP are shown. The firing rate is shown in A, the order parameter in B, the average NDD in C, and the CV of firing rates in D. Colors correspond to the three initial values of average NDD (see legend). The solid lines show the values averaged over 10 realizations and the shaded area represents the error bars. Parameters: 〈W〉(0) = 0.95, σ_f = 0.5 Hz, and P_w = 0.01. Other parameters are specified in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g010

Fig 11 shows the steady state distribution of the firing rates of the original synchronized random network with STDP-only (labeled as initial) and the distributions in the steady states after the network evolved with either STDP+hSP or STDP+SP (P_w = 0.01). The distribution for the synchronized random network with STDP-only is broad and has a peak near its mean value of ≈ 4.2 Hz. In the steady states with both STDP+hSP and STDP+SP, the distributions become narrow with peaks at the corresponding mean value of 4.5 Hz (same as f_T), indicating a significantly lower firing rate mismatch or CV compared to the original network with STDP-only. We added the distribution for the network constructed from scratch using hSP-only to show the reduction in firing rate mismatch in the presence of hSP alone.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Distributions of steady-state firing rates of all N = 400 neurons in the network for different plasticity cases.

The initial distribution of firing rates is that of the synchronized random network with STDP-only. The networks were initialized with 〈W〉₀ = 0.95 and β₀ = 0.075 for all cases but hSP-only, σ_f = 0.5 Hz for all cases, and P_w = 0.01 for the network with STDP+SP. The values are averaged over 10 realizations of the network for each plasticity case.

https://doi.org/10.1371/journal.pcbi.1012261.g011

To demonstrate the enhancement of synchronization when an initially random network with STDP-only progresses through SP iterations for both STDP+hSP and STDP+SP, we consider a synchronized random network with average NDD, β = 0.075, and set the target firing rate f_T = 4.5 Hz as above. We use different values of the maximal probability of weight-dependent pruning, P_w, while keeping the probability of homeostatic addition and pruning constant, P_h = 0.01. Fig 12B shows that the order parameter increases with SP iterations starting from lower values corresponding to the different realizations of the initial synchronized random network with STDP-only for all values of P_w, demonstrating the enhancement of synchronization regardless of P_w. Interestingly, the order parameter, the network-averaged firing rate, and the CV of firing rate evolve similarly and converge to almost equal steady-state values for all considered values of P_w, as shown in Fig 12. However, the evolution of average NDD depends on P_w until the target firing rate is attained by the network, and the steady-state average NDD decreases with the increase in P_w. It is observed because when P_w ≠ 0, the weaker contacts get pruned. A higher P_w causes the pruning of a larger number of weak contacts. The weight-dependent pruning is absent for P_w = 0 (i.e., STDP+hSP), thus the average NDD remains higher than that for P_w ≠ 0. This reduction in average NDD in the synchronized state due to weight-dependent pruning while the network order parameter remains high points to optimization of the network structure such that the contacts that support or promote synchrony are retained while the others that may not contribute to synchronization are eliminated. The convergence of the network-averaged firing rate to the target firing rate depends on the steepness of the logistic curves of the hSP rule (see Fig 1), controlled by the parameter Δf. Less steep logistic curves used in this study may result in slight overshooting (for small values of P_w) or undershooting (for P_w close to 1) of the network-averaged firing rate relative to the target firing rate. Steeper logistic curves result in almost exact convergence of the network-averaged firing rate. Replacing the stochastic SP method with BvOSP gives similar results (see Fig D in the S1 Text).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. The dependence of network dynamics on the maximal probability of weight-dependent pruning, P_w.

Evolution of network-averaged measures as the initially random synchronized network progresses through SP iterations using STDP+SP for the indicated values of P_w. STDP+SP becomes STDP+hSP for P_w = 0. The firing rate is shown in A, the order parameter in B, the average NDD in C, and the CV of firing rates in D. The solid lines show the values averaged over 10 realizations and the shaded area represents the error bars. Parameters: β₀ = 0.075, 〈W₀〉 = 0.95, f_T = 4.5 Hz, and σ_f = 0.5 Hz. Other parameters are specified in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g012

Statistics of synaptic contacts in a synchronized state.

During the transition to a synchronized state, the firing rates of the neurons converge to steady-state values close to the network average. In order to reach the steady-state values, the incoming contacts of the neurons with lower natural firing rates get potentiated and those of the neurons with a higher natural firing rate get depressed on average due to STDP. At the same time, the outgoing contacts of neurons with higher (lower) firing rates tend to get potentiated (depressed) on average, resulting in a decrease in the average incoming synaptic weight of neurons, and an increase in the outgoing weight is observed with an increase in the natural firing rate, as shown in Fig 13A. Correspondingly, the distribution of the weights of individual synaptic contacts in Fig 13B[pink color] shows two peaks at 0 and 1 in the steady synchronized state of a random network with STDP-only.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Average synaptic weight depends on the natural firing rate of neurons and weight distributions differ for different plasticity cases.

A shows the dependence of average incoming and outgoing synaptic weight on the natural firing rate of neurons for a network with STDP-only. B shows the distribution of the synaptic weights of individual contacts for three cases of plasticity: STDP-only, STDP+hSP, and STDP+SP (P_w = 0.01). The values are averaged over 10 realizations of random networks. Other parameters are the same as in Fig 12.

https://doi.org/10.1371/journal.pcbi.1012261.g013

When a random network with STDP-only goes through structural updates with STDP+SP (P_w ≠ 0), the weaker contacts get pruned preferentially, while the existing and the newly added contacts that get potentiated continue to dwell and accumulate over time to raise the firing rates of the neurons to the given target rate. For that reason, the peak in the weight distribution at 0 in Fig 13B reduces considerably and the peak at 1 becomes larger than that for the synchronized random network with STDP-only. When P_w = 0, i.e., for STDP+hSP, both strong and weak contacts may survive due to the absence of preferential pruning of weaker contacts and accumulate over time until the target firing rate is achieved, which results in peaks at 1 and 0 in the synaptic weight distribution in Fig 13B. We observe the same using BvOSP in place of stochastic SP, as shown in Fig E in S1 Text.

Owing to the higher (lower) average incoming (outgoing) synaptic weight, the neurons with lower natural firing rates possess more incoming contacts and fewer outgoing contacts when P_w ≠ 0, as indicated by Fig 14A, since the weaker contacts get pruned. The opposite holds for the neurons with higher natural firing rates. The contrasting dependence of in- and out- NDDs on the natural firing rate results in a negative correlation between those as implicated by their joint probability in Fig 14C. When P_w = 0, the NDDs do not follow the dependence of synaptic weight on firing rate since weaker contacts are not preferentially pruned. Nevertheless, the neurons with lower natural firing rates require more contacts to reach the target firing rate, whereas those with higher rates require less. Thus, we see a minor dependence of in-NDD on the natural firing rate and none for out-NDD in Fig 14B. The joint probability distribution of in- and out- NDDs in Fig 14D shows a positive correlation between the two, which arises due to the existence of neurons on the boundary of the network that develop fewer incoming and outgoing contacts due to the distance-dependent contact formation. In contrast, the ones away from the boundary can develop a higher number of both. The neurons on the network boundary that possess lower natural firing rates build fewer but stronger contacts during synchronization. Replacing the stochastic SP with BvOSP giver similar results for STDP+hSP but different for STDP+SP, presented in Fig F in S1 Text.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 14. in- and out- NDDs depend on the natural firing rates of the neurons.

A and B show the dependence of in- and out- NDDs of networks with STDP+SP (P_w = 0.01) and STDP+hSP on the natural firing rates of the neurons in the steady synchronized states, and C and D show their joint probability distribution, P(in- NDD, out- NDD). The values are averaged over 10 realizations of initial random networks. Other parameters are the same as in Fig 12.

https://doi.org/10.1371/journal.pcbi.1012261.g014

Fig 15 compares NDD distributions in steady synchronized states for STDP+hSP and STDP+SP, i.e., P_w = 0 and P_w = 0.01, respectively. For P_w = 0, both in- and out-NDD distributions have a similar shape and possess the same mode. The accumulation of weaker contacts due to the absence of their preferential pruning results in a shift of the in- and out-NDDs to higher values than the initial distributions. On the contrary, when weight-dependent pruning is on, significantly different distributions of in- and out-NDDs arise. The in-NDD distribution is shifted towards smaller values compared to the initial because of the pruning of weaker contacts. The out-NDD distribution becomes broader and uniform at small values, indicating that neurons possess a wider range of degrees and many have a small number of outgoing contacts. We observe the same using BvOSP in place of stochastic SP, as shown in Fig G in S1 Text.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 15. Marginal distributions of in- and out-NDDs for the networks with STDP+hSP and STDP+SP (P_w = 0.01) in steady synchronized state.

A shows the in-NDD distributions and B shows the out-NDD distributions. The filled histogram shows initial degree distributions with β₀ = 0.075. The values are averaged over 10 realizations of initial random networks. Other parameters are the same as in Fig 12.

https://doi.org/10.1371/journal.pcbi.1012261.g015

The dependence of the average incoming and outgoing synaptic weight of the neurons on their natural firing rate can be attributed to the potentiation of contacts going from neurons with a higher firing rate to those with a lower firing rate and the depression of the ones going the opposite way. This phenomenon is evident from Fig 16A, which shows the dependence of the average lifetime of the contacts on the difference between the natural firing rates of the paired neurons, δf = f_i − f_j, in a synchronized network in the presence of weight-dependent pruning, i.e., STDP+SP (P_w = 0.01). Here f_i is the natural firing rate of the postsynaptic neuron and f_j is that of the presynaptic neuron. The average lifetime of contacts that appear between a given pair of neurons is calculated after the network settles in a steady synchronized state with 500 SP iterations. When the postsynaptic neurons fire at a higher rate than the presynaptic (δf > 0), most contacts become short-lived owing to their depression. In contrast, the contacts between neurons with δf < 0, tend to live for longer because of potentiation. As a result, more contacts exist between neurons with δf < 0 in the steady synchronized state, as indicated by the distribution of δf for connected pairs of neurons in Fig 16B. The peak in distribution close to 0 indicates that neurons with similar natural firing rates are more likely to remain connected in a synchronized state.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 16. The survival of synaptic contacts is subject to the difference between the firing rates of the neurons connected by them.

The dependence of the average lifetime of the synaptic contacts (A) and the number of contacts (B) in the steady synchronized state of the network with STDP+SP on the difference between the firing rates of the post- and presynaptic neurons, δf. δf > 0 when the postsynaptic neuron has a higher firing rate than the presynaptic and δf < 0 when the opposite holds. The values are averaged over 10 realizations of initial random networks and the error bars show the corresponding standard deviation. Parameters are P_w = 0.01, 〈W〉 = 0.95, σ_f = 0.5 Hz, and β(0) = 0.075. P_w = 0.1 and 1 give similar results.

https://doi.org/10.1371/journal.pcbi.1012261.g016

Next, we examine the emergence of correlation between the node degrees of connected neurons in a synchronized network with STDP+SP (P_w = 0.01). The standard measure of node degree correlations is the Pearson correlation coefficient (Eq 18), which also serves as a degree assortativity measure for complex networks [56, 87, 88]. For a directed graph, this measure gives four values assessing correlations between (in-in), (out-out), (in-out), and (out-in) node degrees [88, 90], where the first degree-type corresponds to the presynaptic neuron and the second one corresponds to its postsynaptic partner. Fig 17 shows the Pearson correlation coefficient values versus the diversity parameter, σ_f. The correlation coefficient is averaged over ten realizations of networks for each value of σ_f. A positive (negative) correlation value implies that the network is assortative (disassortative). We observe that for identical neurons or smaller diversity (σ_f < 0.2), the networks become assortative for all degree types of the pre- and post-synaptic neurons. For larger diversity (σ_f > 0.2), a mixture of assortativity and disassortativity emerges as reported for other networks [89]. On the one hand, the (in-in) and (out-out) correlations are positive and increase with an increase in σ_f, implying that the networks are in-assortative and out-assortative [90] and the networks with a higher σ_f are more assortative. On the other hand, the negative (in-out) and (out-in) correlations and their decrease to more negative values with an increase in σ_f imply that in that respect, the networks become increasingly disassortative [89].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 17. Network degree-(dis)assortativity depends on the diversity parameter, σ_f.

The Pearson correlation calculated between the NDDs of the pre- and post-synaptic neurons, labeled as (NDD-type of pre, NDD-type of post) is shown. The values are averaged over ten realizations of initial random networks and the error bars show the standard deviation for each value of σ_f. Parameters: P_w = 0.01, β₀ = 0.075, and 〈W〉 = 0.95.

https://doi.org/10.1371/journal.pcbi.1012261.g017

The emergence of non-zero node-degree correlations can be attributed to two factors: first, the spatial arrangement of neurons on a square lattice accompanied by distance-dependent formation of contacts, and second, the existence of more synaptic contacts between neurons with similar natural firing rates in a synchronized state, as suggested by the peak around 0 in Fig 16B. The boundary effect arises due to the availability of fewer neurons close to the network edges, letting the neurons on the edges build fewer incoming and outgoing contacts. This can be countered by firing-rate-dependent incoming contact formation but it prevails for the outgoing contacts as those are not added based on the firing rate of the presynaptic neurons. Consequently, the out-degree of neurons on the edges is smaller and that of neurons away from the edges is larger; combined with the likelihood of forming contacts with more nearby neurons, it results in higher positive values of (out-out) correlation regardless of the diversity parameter. The firing-rate-dependent incoming contact formation makes the neurons for small σ_f < 0.2 have similar degrees, resulting in positive (in-in), (in-out), and (out-in) correlations. However, the degrees show increasing dependence on the natural firing rates as σ_f increases due to the weight-dependent pruning. Thus, when neurons with similar firing rates get connected, they are more likely to have similar degrees for a large σ_f, resulting in a higher (in-in) correlation. At intermediate values of σ_f, the dependence of both in- and out-NDDs on natural firing rate is weaker than that for a larger σ_f, resulting in a lower (in-in) correlation. Since the in- and out- degrees depend oppositely on the natural firing rates [Fig 14A], the (in-out) and (out-in) correlations become increasingly negative for larger σ_f. Without weight-dependent pruning (P_w = 0), the networks show no significant node degree correlations, i.e., the Pearson correlation coefficient is not significantly different from 0.

Desynchronized states with a combination of structural and weight plasticity

Steady synchronized states are obtained when the initial average NDD and synaptic weight are sufficiently high [cf. Figs 3 and 6]. In our model, a steady synchronized state is reached when the network-averaged firing rate reaches f_T, bringing the average NDD to its steady value. A desynchronized state of a random network with STDP-only is obtained when the initial synaptic weights are small. In desynchronized states, the neurons remain weakly coupled due to the persistent depression of most of the synaptic weights. Consequently, the network-averaged firing rate may shift to larger values than the average natural firing rate, f₀, but may not reach f_T if it is much higher than f₀. If a desynchronized random network with STDP-only undergoes SP iterations as described in ‘Parameters and implementation’ section, the initial structural updates only involve the addition process because of lower firing rates of the neurons. The neurons may continue to develop contacts while never reaching f_T until the network reaches all-to-all connectivity. The pruning process slows down the increase in average NDD and may halt it by balancing the number of added contacts. Consequently, a steady desynchronized state of a network with STDP+SP or STDP+hSP is reached when the pruning process balances the addition, bringing the average NDD to a steady value. The rates at which contacts are added and pruned may, therefore, have a significant impact on the network density in the steady desynchronized state.

We study the effect of the maximal probability of weight-dependent pruning, P_w, on the steady-state average NDD of the network in a desynchronized state. SP updates were implemented until the average NDD converged with the relative accuracy of 0.1%. We set f_T = 4.5 Hz as for the synchronized networks in the above sections so that the target firing rate is higher than the natural firing rates of all the neurons in the network. Fig 18A shows that the network-averaged firing rate may increase in the desynchronized state depending on P_w but fail to reach f_T since it is much higher than f₀ = 3 Hz. Additionally, the homeostatic pruning remains negligible. Therefore, Fig 18C shows that in the absence of weight-dependent pruning, P_w = 0 (STDP+hSP), the neurons continue to build contacts and the network becomes densely connected. With weight-dependent pruning, i.e., P_w ≠ 0, the network loses weak contacts, reducing the steady-state average NDD. This effect increases with increasing P_w.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 18. Evolution of network-averaged measures in the desynchronized state.

Networks with STDP+hSP (P_w = 0) and STDP+SP (P_w ≠ 0) for the indicated values of P_w are shown. A shows the firing rate, B shows the order parameter, and C shows the average NDD. The solid lines show the values averaged over 10 realizations and the shaded area represents the error bars. Parameters: β₀ = 0.075, 〈W₀〉 = 0.05, and σ_f = 0.5 Hz. Other parameters are specified in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g018

Fig 19 shows the decrease in steady state values of the average NDD in the desynchronized state with an increase in P_w when P_h = 0.01 for f_T = 4.5 and 5.5 Hz. We also present the average NDD in a steady synchronized state of the network for the same values of f_T in Fig 19 for comparison. For sufficiently high values of P_w/P_h, the desynchronized network has a significantly lower average NDD than the synchronized network. For example, for f_T = 4.5 Hz, P_w = 1 and P_h = 0.01, the desynchronized network has β ≈ 0.023 whereas the synchronized network has β ≈ 0.062. It is consistent with the results reported in our previous study that used a simple phase oscillator model for neurons [53]. On the contrary, when P_w and P_h are comparable, the desynchronized network can be significantly denser than the synchronized network. For example, for P_w = P_h = 0.01, the desynchronized network has β ≈ 0.5 whereas the synchronized network has β ≈ 0.07, for f_T = 4.5 Hz.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 19. Steady-state average NDD depends on the probabilities of homeostatic SP and weight-dependent pruning.

The value of the steady-state average NDD, β, versus the ratio of the maximal probability of weight-dependent pruning, P_w, to the probability of homeostatic addition and pruning, P_h, for the indicated values of the target firing rate in desynchronized and synchronized states. The initial value of average NDD is β₀ = 0.075; the initial average weight is 〈W₀〉 = 0.05 for desynchronized states, and 〈W₀〉 = 0.8 for synchronized states.

https://doi.org/10.1371/journal.pcbi.1012261.g019

Effect of desynchronizing stimulation

The synchronized spiking of neurons in our network may be used as a simple model representing pathological states of certain brain areas in neurological disorders, such as PD [97], epilepsy [98], and tinnitus [99, 100]. A number of stimulation methods have been designed to counter abnormal synchrony, induce long-term desynchronization, and restore normal spiking pattern, for instance, in PD [24, 32, 33].

The raster plots in Fig 20 exemplify the effect of desynchronizing stimulation, UMRS, on a synchronized network with STDP-only. The stimulation disrupts the synchronized firing (middle panel), causing a desynchronization and moving the network into the basin of attraction of a desynchronized state. Accordingly, after removing the stimulation, the desynchrony persists, as shown in the right-most panel.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 20. Effect of UMRS stimulation on a synchronized network with STDP-only.

The raster plot on the left shows a synchronized state, the middle raster is obtained during the desynchronizing stimulation period, and the raster on the right shows the desynchronized state achieved post-stimulation. Stimulation parameters: a_s = 0.4, F_s = 50 Hz, and duration = 2 minutes. Other parameters are specified in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g020

As demonstrated in previous sections, the type of plasticity affects the spontaneous dynamics of the networks. The synaptic reorganization that occurs when STDP is combined with SP (or hSP) enables the network to become more strongly synchronized than the random network with STDP-only (see Figs 9, 10, and 12). To study the effect of UMRS on the synchronized networks, we consider two random networks with STDP-only, one with STDP+SP (P_w = 0.01), and one with STDP+hSP. For given parameters, σ_f = 0.5 Hz, P_w = P_h = 0.01, and f_T = 4.5 Hz, the network with STDP+SP settles in a synchronized state with β ≈ 0.07 starting from any value of β₀ as seen in Fig 10. However, a random network with STDP-only with the same parameters and β₀ = 0.07 does not converge to a synchronized state [Figs 3 and 6]. Thus, we consider a random network with STDP-only having β₀ = 0.08 to keep the average NDD of the networks with and without SP very similar and prepare both in a synchronized state. This network, however, has a lower order parameter, 0.7, and network-averaged firing rate, ≈ 4.2 Hz, than the network with STDP+SP, with 0.9 and ≈ 4.5 Hz, respectively. Thus, we also consider a random network with STDP-only having β₀ = 0.105 with an order parameter 0.86 and network-averaged firing rate 4.5 Hz, close to the network with STDP+SP. The network with STDP+hSP settles with β = 0.106 and the order parameter and average firing rate similar to those of the network with STDP+SP. To highlight the role of network structure that emerges in the presence of SP or hSP on the robustness of networks against stimulation, we halt the structural updates during and after stimulation.

Fig 21 compares the intensity and frequency of stimulus needed to induce desynchronization in the networks considered. The maps show two regions of the combinations of amplitude and frequency of the stimulus current administered for a two-minute window. The blue regions show those combinations that successfully desynchronize the initially synchronized networks, while the red regions show those that do not. The extension of the blue region in Fig 21A to lower values of amplitude and frequency than Fig 21C indicates that when the average NDDs of the network with STDP-only and the one with STDP+SP are close, the one with STDP-only requires much lower amplitude and frequency of stimulus to get desynchronized. The blue and red regions in Fig 21B, 21C and 21D show negligible differences, implying that the network with STDP-only as well as the network with STDP+hSP require much higher average NDD to be as robust against stimulation as a network that evolved with STDP+SP.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 21. Desynchronization of synchronized networks after stimulation using UMRS.

A and B show networks that evolved with STDP-only, C shows the network with STDP+SP (P_w = 0.01), and D shows the network with STDP+hSP (P_w = 0). The colors show the order parameter of the networks five hours after a two-minute stimulation window for given combinations of stimulus amplitude and frequency. Red indicates that the network returns to a synchronized state after stimulation, while blue indicates that it settles in a desynchronized state. The stimulus parameters are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g021

Fig 22 illustrates the effect of UMRS stimulus by showing an example of the time evolution of the network-averaged firing rate and the order parameter of the networks considered. All four synchronized networks initially have higher network-averaged firing rates and order parameters. During stimulation, the firing rate increases while the order parameter decreases for all the networks. However, post-stimulation, only the random network with STDP-only and β₀ = 0.08 gets desynchronized as implied by its low order parameter after stimulation, while the other networks relax back to their synchronized states. This is expected since the random network with STDP-only with β₀ = 0.08 is initially less strongly synchronized as indicated by its lower order parameter compared to the other networks. The networks with STDP+SP and STDP+hSP, and the random network with STDP-only with β₀ = 0.105 have similar levels of synchrony and thus, respond similarly to the desynchronizing stimulation.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 22. An example of the effect of UMRS stimulus on the synchronized networks with specified plasticity cases.

The network-averaged firing rate and the order parameter of the network are shown in A and B, respectively. The shaded area shows the stimulation period. P_w = 0 for STDP+hSP and P_w = 0.01 for STDP+SP. Stimulation parameters: a_s = 0.221, F_s = 51 Hz, and duration = 2 minutes. Other parameters are specified in Table 1.

https://doi.org/10.1371/journal.pcbi.1012261.g022

Stochastic model of structural plasticity

Different implementations of adaptive neuronal networks with synaptic plasticity, both weight and structural plasticity, have been used to study either the properties of emergent network structures in the equilibrium [46, 47, 106] or the resulting synchronization dynamics [20, 50–53]. Synchronization is an emergent phenomenon arising from the interaction of the nodes and the structure of a network. Thus, it is essential to capture the complex interactions between neuronal dynamics and synaptic plasticity during transitions to and in steady (de)synchronized states in plastic neuronal networks. Previously, synchronization was studied using simpler models of SP with phase oscillators that preserved the number of contacts and preferentially connected the oscillators that either had similar average frequencies [51] or were more out-of-phase [50], which led to stronger clustered or global synchrony, respectively. Recently, synchronization was studied with detailed neuron models combined with synaptic weight and SP models [20, 52, 53]. However, these studies either did not capture the interaction of synaptic weights and structural change by excluding weight-dependent pruning [20, 52] or imposed homeostatic constraints on the node degrees rather than employing an average activity-based hSP rule [53]. In the present study, we captured the interactions of neuronal dynamics, synaptic weights, and structural changes while maintaining distinct timescales of the three by employing STDP that connected neuronal spiking and synaptic weight change, hSP that connected average neuronal activity (firing rate) and structural change, and weight-dependent pruning that captured the interaction of synaptic weight and structural dynamics.

A neuroscientifically informed SP model, BvOSP [61], has been successfully used in many studies, for example, to generate networks from unconnected neurons and study their reorganization [49, 61, 83]. Its algorithm, however, is computationally costly to execute and therefore may limit the extent to which the analysis of network dynamics may be conducted. Our stochastic SP method, on the other hand, abstracts from the details of the spine and bouton genesis, unlike BvOSP, and directly builds contacts between neurons, allowing for faster computational implementation. We combined it with STDP-driven weight dynamics, unlike most previous studies that employed BvOSP [49, 82, 84]. Our stochastic SP method bears some similarities with BvOSP but differs in many regards. BvOSP permits multiple synaptic contacts between any given pair of pre- and post-synaptic partners, but in our stochastic SP, for simplicity, we assumed that a single contact accounts for the overall effect of the presynaptic neuron on its postsynaptic partner, although it can easily be extended to include multiple contacts. BvOSP determines the numbers of pre- and post- synaptic partners for all neurons due to the explicit dependence of birth and death of both axonal and dendritic elements on average neuronal activity, whereas the stochastic SP method only determines the number of presynaptic partners based on a neuron’s activity level, not the number of postsynaptic partners. Our model of SP successfully reproduced the results obtained with BvOSP in this study.

Networks with STDP-only may settle in steady states with different levels of synchronization depending on the initial conditions

To understand how each type of plasticity governs the network dynamics, we first studied the synchronized states of homogeneous and heterogeneous networks consisting of identical and non-identical neurons, respectively, with STDP-only. In this context, a perfectly synchronized state is defined by the coincident firing of the entire network. Interestingly, the networks attained fully synchronized states for intermediate values of the average node degree and desynchronized states for lower values, while remaining partially synchronized for higher values of the average node degree. Higher values of the average node degree may cause cluster states. It indicates that high connectivity does not necessarily favor the synchronized activity of neuronal networks with STDP-only, in agreement with a previous study [107]. This is also in agreement with pre-clinical evidence. For instance, in Parkinsonian mice, an increase in cortico-STN coherence was accompanied by a substantial (50%-75%) decrease in the number of cortico-STN synapses [108].

SP promotes and enhances synchronization

We examined how the heterogeneous networks evolved with hSP-only. hSP has been used to automatically generate networks that evolve to attain a target activity level [49, 81, 82]. We generated networks using BvOSP and stochastic SP separately from completely unconnected neurons that built contacts in order to reach a given target firing rate, f_T. The neurons with a natural firing rate below f_T could build new incoming contacts while the ones with higher rates could not. Consequently, a sufficiently large f_T led to a sufficiently large average node degree that enabled the network to settle in a synchronized state while a lower target allowed only for desynchronized or partially synchronized states. Our stochastic SP method, thus, offers a simpler way to generate networks with the desired level of activity and synchrony.

Next, we developed networks from unconnected neurons using STDP+SP (P_w > 0) and STDP+hSP (P_w = 0) and compared their transition to synchronized states as a function of network density with networks generated using hSP-only and the random networks with STDP-only. The network developed from scratch using hSP-only required the highest average node degree to achieve synchrony, while the network with STDP-only could get synchronized with a much smaller average node degree. The smallest average node degree was required by the networks with STDP+SP and STDP+hSP to reach a synchronized state.

We studied the synchronization dynamics of heterogeneous networks with a combination of STDP and SP. To this end, we considered random networks in synchronized states with STDP-only and applied structural updates (SP iterations), as outlined in the ‘Parameters and implementation’ section. As the network progressed through SP iterations, the neurons became increasingly more identical by adding or losing contacts as their firing rates converged to a single target rate with both SP methods (stochastic and BvOSP) regardless of weight-dependent pruning, making way for enhanced network synchronization for any value of the probability of weight-dependent pruning, i.e., P_w ∈ [0, 1]. Nevertheless, P_w determined the final average node degree for a given f_T such that a higher P_w resulted in a sparser network.

SP optimizes the network structure by preserving the synaptic contacts that support synchronization

The average synaptic weight of both incoming and outgoing contacts displayed a strong dependence on their natural firing rate in the presence of STDP. Particularly, the average incoming weight decreased with an increase in the natural firing rate, while the average outgoing weight increased since the contacts directed from neurons with a higher natural firing rate to those with a lower rate got potentiated on average while the others got depressed. As the initially random network in the synchronized state underwent structural updates in the presence of weight-dependent pruning, the potentiated contacts survived for longer times, while the depressed ones eventually got pruned. As a consequence, the structure of the network in the presence of SP underwent a reorganization governed by the natural firing rates of the neurons.

With the stochastic SP method, the above factors resulted in a strong relationship between the natural firing rate of the neurons and their in- and out- degrees, such that the neurons with higher (lower) natural firing rates had more (fewer) outgoing and fewer (more) incoming contacts, suggesting that effectively the faster neurons (those projecting densely onto others) drove the slower ones (those receiving more inputs). This finding matches with the experimental observation that in the suprachiasmatic nucleus in the brain, the neurons in the core project heavily onto the neurons in its shell while receiving sparse inputs from the shell [109, 110] and drive synchrony by leading the phase and entraining the neurons in the shell [111]. With BvOSP (P_w > 0), the dependence of out-degree on the natural firing rate of the neurons in a steady state was different than that with the stochastic SP model since BvOSP identically controlled both in- and out-degrees via firing rate-dependent sprouting of both dendritic and axonal elements, such that the neurons with a natural firing rate close to the target formed fewer dendritic and axonal elements, while our stochastic SP did not directly constrain the out-degree. With BvOSP, the out-degree first increased with an increase in the natural firing rate, reached a maximum, and dropped to a small value near f_T, unlike with stochastic SP where the out-degree monotonously increased with the natural firing rate. With P_w = 0, both methods produced no degree-frequency correlation.

The removal of weak contacts due to weight-dependent pruning could be viewed as a possible way of optimizing the network structure for synchrony. In a theoretical study, robust synchronization was suggested to benefit from the removal of links [107]. We previously observed such optimization of network structure in a study with networks of oscillators, where the networks that evolved with a combination of STDP and SP got synchronized for significantly smaller average degrees compared to random networks with STDP alone [53].

The degree-frequency correlation above offers insight into the co-evolution of neuronal activity and structure as the activity level of neurons governs the degrees of the network, and the degrees, in turn, alter the level of activity of the neurons. Degree-frequency correlation has been observed in adaptive networks of oscillators where the correlation emerged as a result of either preserving contacts between out-of-phase oscillators while rewiring those between more in-phase ones [50] or due to the dependence of the synaptic weight on the frequency difference between the connected oscillators, accompanied by weight-dependent pruning [53], similar to the present study.

A mixture of assortativity emerges in synchronized networks with STDP+SP

In a steady synchronized state of a network with STDP+SP (P_w ≠ 0), more neurons that remained connected had similar natural firing rates. Since the in- and out-degrees of a neuron depended oppositely on its natural firing rate for a sufficiently large diversity in natural firing rates, the networks portrayed a mixture of degree assortativity by becoming in- and out-assortative while remaining disassortative with respect to the opposite degree types of the connected neurons. The random networks we considered were found to be non-assortative, i.e., Pearson correlation coefficient = 0, similar to previously reported findings [56, 112]. For a weighted graph, the weighted degree should be used to calculate the assortativity [91, 113]; however, for consistency and simplicity, we reported only node degree assortativity. Large-scale networks in the human cerebral cortex were found to be assortative using undirected graph analysis, and the same networks when randomized became non-assortative [114]. Assortative neural networks are also suggested to be more robust in information storage against noise [115]. On the one hand, the assortative nature makes a network robust against node failure or targeted attack and increases the speed of the information transfer [56, 87, 88]. On the other hand, it makes networks more unstable or vulnerable to external perturbation [116].

Desynchronized networks with combinations of weight and structural plasticity

Next, we studied the transition of the network with STDP+SP to a desynchronized state for various values of the weight-dependent pruning probability, P_w ∈ [0, 1]. The average node degree decreased with an increase in P_w when the homeostatic probability of addition and pruning, P_h, remained fixed. A comparison of the node degrees in the steady synchronized and desynchronized states revealed that depending on the ratio of P_w to P_h, the network in the two states could have significantly different average node degrees. Particularly, comparable values of P_w and P_h made the desynchronized network considerably denser than the synchronized network, while the larger ratios, P_w/P_h, could make the synchronized network much denser than the desynchronized one. In our previous study with simple phase oscillator networks, where the weight-dependent pruning was 100 times faster than homeostatic SP, the desynchronized networks with STDP+SP ended up being considerably sparser than the synchronized network.

In summary, we found that the network structure is strongly related to its dynamical state, and the networks can spontaneously enter pathological or physiological states depending on initial conditions. Future studies should also take into account the impact of stimulation on both abnormal and physiological connectivity, e.g., to improve restoration of segregated activity and related connectivity. By the same token, plasticity mechanisms are crucial for physiological mechanisms. For instance, studies on networks of the interaction of organ systems have demonstrated a similar phenomenon where the network structure is correlated with specific physiologic function such that a change in the interaction network structure is associated with a change in the function [117–119]. Furthermore, the brain wave network interaction and its plasticity are shown to be crucial in creating physiological states and distinct interaction patterns [119, 120]. Accordingly, beyond the specifically desynchronization-oriented approach mentioned above, our model aims to provide a testbed for further development of stimulation methods that cause long-lasting stimulation effects by reshaping connectivity and related neuronal activity in the presence of different plasticity mechanisms.

SP makes synchronized networks more robust against desynchronizing stimulation

The structure of a network in a synchronized state is critical in determining its robustness against external perturbation. Thus, studies of network structure properties hold importance in the design of therapeutic stimulation methods that have the potential to induce long-lasting relief from the severe effects of pathological synchrony in particular brain areas in certain neurological disorders. Some of the theoretically developed versions of CR stimulation successfully induced long-lasting desynchronizing and/or therapeutic effects in pre-clinical and clinical studies in PD and clinical studies in tinnitus patients as well as in animal models of epilepsy and binge alcohol drinking [71–73, 75–78, 121–123]. RR stimulation [21, 22] and other multichannel stimulation techniques [124, 125] were theoretically developed to induce long-term desynchronizing effects. We here introduced UMRS which shares the temporal randomness of RR stimulation, combined with the spatially fixed multi-site setup of several previous multichannel stimulation techniques [124, 125]. It reliably induced long-term desynchronization in our model networks. While the assortative nature of the networks that evolve with the combination of STDP and SP suggests that they should be more vulnerable to stimulation, the optimization of the network structure for stronger synchrony due to SP suggests that such networks should require stronger stimulation than the random networks with similar average node degrees. The latter outweighed the former in our study since we found that the random network with a similar average node degree as the one that evolved with SP required significantly lower stimulation to get desynchronized. A variety of different desynchronizing stimulation techniques were computationally developed to achieve optimal long-term desynchronization for specific conditions and constraints [21, 22, 24, 33, 126–128]. Analogously, given the slow SP time scale, desynchronizing stimulation techniques can be further developed using network models with STDP and SP with acceptable computation time requirements, hence enabling thorough computational analyses (see, e.g., Refs. [47, 53, 129]). Minimal network models, as used in this study, may enable predictions for the numerical study of more complex, biologically more realistic networks [52, 130–134]. Accordingly, beyond the specifically desynchronization-oriented approach mentioned above, our model aims to provide a testbed for further development of stimulation methods that cause long-lasting stimulation effects by reshaping connectivity and related neuronal activity in the presence of different plasticity mechanisms.