Modularity and multitasking in neuro-memristive reservoir networks

All results are obtained from simulations based on physical and biologically-inspired networks, each modelled with the same memristive synapses. We first compared a self-assembled NWN with three other physically-motivated networks with different topologies: sparse crossbar array, a randomly-rewired NWN (hereafter referred to as a random network) and the C. elegans structural connectome. A crossbar array, typically used in memristor devices [41, 42], has a bipartite network structure. The random network represents an NWN artificially reconstructed to remove any complex network structure, while C. elegans is a biological neural network. The average degree (⟨k⟩ ≈ 13), number of nodes (between 277–300) and number of edges (between 1800–2100) were kept as similar as possible to the C. elegans network, to best compare the effect of each network's structure on its function. Graphical representations of the networks are shown in figure 1(A).

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Figure 1(B) shows four graph theory metrics used to measure structural properties of each network: network diameter (which is also the shortest path length from source to drain electrode), modularity (Q), average clustering coefficient (global clustering) and small worldness (SWP). The NWN (red) exhibits the highest modularity, longest diameter, highest average clustering coefficient and highest small worldness. To investigate the structure–function relationship of these networks, all networks were simulated with the same voltage-controlled memristive edge–junction model, based on conductive filament formation and annihilation (see methods). Figure 1(C) compares the performance of each network on two different benchmark tasks: NLT and MC. Clearly, the NWN (red) performs relatively poorly on both tasks at low voltages (<1.75 V), but comparably well, or even better at V ≲ 1.75 V. When the NWN is randomly rewired (green), it has significantly lower modularity, diameter, average clustering coefficient and small worldness, yet its performance on the NLT and MC tasks is generally superior across all voltages, except at V ≲ 1.75 V for the NLT task.

The C. elegans network (yellow) falls between the NWN and its randomly-rewired counterpart in diameter, clustering, modularity and small worldness. It performs comparably to the random network on NLT and worse on the MC task. Compared to the NWN, C. elegans performs better at the NLT task at lower voltages, and comparably at higher voltages, as well as slightly better at lower voltage and significantly worse at higher voltages on the MC task.

Lastly, the sparse crossbar array network (blue) has comparable structural parameters to the random network, with similar diameter, modularity and small worldness, however it shows no clustering at all. The network performs comparably to the random network on the NLT task up to 3 V, above which its performance drops. For the MC task, the sparse crossbar array exhibits a similar trend to the other networks, with performance generally increasing with V. It performs better than C. elegans and the NWN over the range 1–3 V and worse than the random network above 2 V.

Overall, the sample self-assembled NWN does not perform as well as its randomly-rewired counterpart on both NLT and MC tasks at lower voltages. Since the random network shares the same degree distribution, number of nodes and number of edges as the NWN, differences in performance must be due to structural parameters, or placement of the electrodes (we show that the latter is not strongly responsible for performance differences in supplementary figure 1 (https://stacks.iop.org/NCE/1/014003/mmedia)).

Figure 2 qualitatively shows functional differences in how the NWN and the random network perform the NLT task. Functional subgraphs showing junction conductance are presented for each network after 10 s of the input signal for three different voltages (see supplementary movie 1 for sample network activation over 10 s for both NLT and MC tasks). For voltage 1, both networks are in a state of low activation. In this state, no conductance paths are formed between source and drain, however paths are beginning to form from each electrode to its closest neighbouring nodes. NLT accuracy for both networks is comparable. Voltage 2 is chosen such that each network performs best (or close to) on the NLT task. Both networks are clearly activated, with one or more high conductance paths formed between source and drain electrodes, but much of the network still remains inactivate. For voltage 3, each network is highly active after 10 s, but NLT performance decreases from the peak.

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A single path formation between source and drain electrodes (also known as the 'winner-takes-all path' (WTA) [43, 44]) has been previously shown to coincide with a sudden increase in network conductance, and correspondingly, task performance [38, 45, 46]. Due to this sharp increase in conductance, we henceforth refer to the WTA pathway formation as the point of network activation. Here, the voltage required to reach activation is higher for NWNs than for random networks (2 V compared to 0.75 V, respectively). As the diameter and average path lengths of NWNs are higher, more voltage is required for the network to form a pathway between the source and drain electrodes. In contrast, since the random network has a smaller diameter, the distance between source and drain electrodes is shorter, so less voltage is required to activate a conducting path between source and drain.

When we control for path length between source and drain in the NWN, while network activation still occurs at a lower voltage (around 1.25 V) compared to the random network, a noticeable difference in network performance persists; i.e. the NWN performs significantly worse (see supplementary figure 1). The decrease in performance can be attributed to a considerably smaller portion of the NWN being activated, relative to the random network, in which much of the network is activated, even though the path length between source and drain is the same in both networks. This stark difference in performance can be largely attributed to differences in network structural connectivity.

As random networks have relatively low modularity, negligible clustering, and low small worldness (cf figure 1), the possible paths to traverse between source and drain are less restricted and therefore more numerous than in NWNs. As such, conductance can readily spread to the closest connected nodes as voltage increases. This allows for much of the network to be activated in significantly fewer steps. In contrast, NWNs are structurally constrained by a more modular structure, with higher clustering and longer path lengths between different sections (communities) of the network. This structure restricts information flow to the closest neighbours of the activated nodes, requiring more steps (and higher voltage) to reach distant parts of the network.

To further explore the effect of structural parameters (including modularity) on NWNs and their performance on benchmark tasks, we performed a parameter sweep across different network topologies. In the following sections, we explore the effect of varying network density and modularity on NLT and MC task performance.

In the network comparison above, network density was kept constant (with ⟨k⟩ ≈ 13) across the four different types of networks. Here, we explore the change in performance on NLT and MC tasks when the average degree of NWNs is varied. To do so, we constructed networks of ≃300 nodes, with an average degree ranging from 5 (highly sparse) to 286 (highly dense). For more information about network construction, see methods. Importantly, at very high densities we see a vast reduction in small worldness (cf table 2). This was also shown in a quasi-3D model of NWNs [31].

Figure 3 shows a heatmap of average performance across ten networks for each network density (columns) increasing from left to right, with varying input voltage (rows). The top panel shows mean accuracy on the NLT task, and the bottom panel shows mean MC score.

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At low densities, high voltage is required to drive the network into activation (based on Kirchoff's circuit laws). This is due to longer path lengths between source and drain, and fewer parallel paths for the voltage to transverse [46, 47]. As density increases, so too does the number of connections and parallel paths, so that lower voltage is required to activate the network. For the NLT task, this results in higher performance at higher densities and lower voltages. Nonetheless, above a certain density, performance decreases. This is due to too much of the network being activated, similar to the effect observed in the functional subgraphs in figure 2 at high voltage. Figure 4 shows the corresponding functional subgraphs for varying density. At high density, the signal spreads indiscriminately throughout the network and limits NLT accuracy (and MC score, but at higher density—cf figure 3). At low to medium densities, when the network is activated, only sections of the network that are on or near the WTA pathway are activated. Both NLT and MC performance improves dramatically when the WTA is formed. We have shown this to be associated with a first order phase transition [47]. While some parallel or forking pathways may be seen, much of the network remains inactive. The competition for pathway formation ensures a diverse variety of junction dynamics and hence, richer and more separable output features that serve to improve regression to the target waveform in the NLT task [27, 48].

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For the MC task, figure 3 shows a more consistent improvement in performance as density increases, up to a level (around 256–271) above which catastrophic failure occurs. Below this, performance peaks at an average degree of around 97 for a range of voltages (cf figure 4). Why are these results so different from the NLT task? The MC task is inherently different from the NLT task: output features are regressed to a target signal (the input signal delayed by varying amounts) that is random and more rapidly varying than the periodic target signal used in NLT. Thus, both tasks involve different timescales and use different reservoir resources. MC relies mostly on the reservoir's fading memory due to the recurrent connections, whereas NLT relies mostly on the resistive memory of junctions, which has slower dynamics [27]. Thus, as density increases with the number of junctions, it becomes increasingly more difficult for the reservoir to remember high frequency components of the input signal as the reservoir resources become overwhelmingly dominated by the lower frequency dynamics of resistive memory.

To test the hypothesis that modularity enables multitasking, we segmented the original NWN into two separate modules (see methods for the full network construction process). We then randomly rewired edges from the two separate modules in six steps, decreasing modularity from two, fully-separated modules to a network with two highly segregated modules, to a network with two highly integrated modules (similar to the random network in section 1). We repeated this process for ten total NWN realisations. The panels beneath figures 5(A) and (B) show the different modularity values, from fully integrated (left) to fully segregated and separated (right), where the modules are not connected at all.

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We performed the NLT and MC tasks simultaneously in each of the two network modules. Figure 5(A) plots the mean NLT accuracy and mean MC score as a function of input voltage for varying levels of modularity. Figure 5(B) plots the corresponding normalised multitasking score for both tasks (see methods for details on how this score is calculated). Figure 5(C) compares the normalised multitasking score for the three cases of a fully integrated network, a highly segregated network and separated network modules (respectively corresponding to the first and last two columns of figure 5(B)).

From figure 5(A), the NLT task performs best when the network modules are fully separated. Although the highly segregated network performs comparably well, performance drops abruptly from fully separated to highly integrated, with the most integrated networks generally performing significantly worse at NLT. Contrastingly, for the MC task, performance decreases catastrophically for fully separated modules compared to highly segregated networks. However, integrated networks slightly outperform segregated networks, particularly at higher voltages.

When considering multitasking performance (figure 5(B)), segregated networks outperform integrated networks at almost all voltages (except for 0.5 V). A sharp drop in performance is evident in fully separated network modules compared to highly segregated networks. Indeed, as figure 5(C) shows, fully separated network modules, with no connections, perform considerably worse at multitasking than either highly segregated or integrated networks. The overall best performance on multitasking is found for segregated networks.

To test if the modular structure is directly responsible for this result, we swapped the source electrode from module 2 with the drain electrode of module 1, such that both sources are in one module, and both drains in the other. Under this configuration, performance drops drastically for the segregated networks, (see supplementary figure 2). This confirms that for modular structure to be useful for multitasking, each task must be allocated to a dedicated module. Moreover, our results suggest that some level of crosstalk between modules is necessary to achieve optimal multitasking performance.

To better understand why segregated networks perform better than integrated networks at multitasking, we investigated NWN activation and functional sub-graphs for different voltages at t = 10 s (see supplementary materials for the full time-series video). Figure 6 compares multitasking performance for segregated (green) and integrated (red) networks, highlighting three key voltages to visualise network activation. These three voltages reflect different states of the networks, the same as in figure 2. The integrated network performs best at a lower voltage, V = 0.5 V, which can be attributed to paths being able to form more readily than in the segregated network. At this voltage level, there is negligible interference between the two tasks, as not much of the integrated network is active, leaving much of the network's resources untouched. At higher voltages, when much of the network is activated, a larger amount of overlap between the two tasks is evident, causing interference, crosstalk and more competition for the same resources. In contrast, the segregated network requires higher voltage for enough of the network to be activated to perform multitasking well. Since the two modules are highly segregated, even at 5 V there is insufficient interference between the tasks to impact performance significantly. Furthermore, different resources are largely being employed for each task, with minimal overlap. A reduction in multitasking performance is only evident at 10 V, at which point most of network is activated, and many more of the same resources are being shared.

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What does interference between the two tasks in such cases look like? When both tasks share too many of the same resources, they are not equally affected by the resulting interference. Performance on the NLT task drastically decreases, while MC remains relatively unaffected. These changes in performance may be attributed to the difference in timescales between the two input signals, as the NLT task relies on a slower timescale while the MC task fluctuates more quickly. A slower timescale (NLT) enables conductive junction filaments to form/decay more consistently, whereas rapid random fluctuations (MC) does not allow for this. As such, even with only a few connections between modules, the MC task acts as a source of increased noise to the junctions in the NLT module, creating instability in filament formation and decay. This is borne out when controlling the voltage of the NLT task while increasing the MC voltage (cf supplementary figure 3). As MC voltage increases, NLT accuracy decreases sharply even if the NLT input voltage is kept constant. Contrastingly, for the MC task, the NLT input acts as a slower modulating signal, increasing MC performance. Analysis of constant voltages further supports the notion that the MC task hinders performance on NLT at higher voltages due to increased crosstalk, and the NLT task boosts performance on the MC task at higher voltages.

Finally, based on the density results showing both NLT and MC utilising different timescales, we wanted to ensure that multitasking results were not improved simply because each task had access to more network resources (i.e. 300 nodes instead of 150 within their own module). Supplementary figure 4 shows that access to more resources is not enough to improve MC to the same levels we see in multitasking, and that NLT acts as a modulating signal to increase MC performance.

Overall, these results suggest that multitasking is most effective in highly modular networks that allow some level of crosstalk between the segregated modules. This, in turn, enables some optimal resource sharing yet limits competition for resources between the modules.

For the two RC benchmark tasks considered here, NLT and MC, we found that NWNs generally require higher voltage or density (i.e. average degree) to perform comparably to random networks, due to their higher modularity, average clustering and small worldness. Our results indicate that these structural properties play an important role in trafficking network information flow, preventing information from spreading indiscriminately throughout the network.

In previous studies, the NLT waveform regression task was tested on atomic switch NWNs both experimentally [21] and in simulation [36]. To perform the NLT task, Demis and colleagues [21] input a sine wave into a NWN hardware device and linearly regressed multiple electrode readouts to a square wave with accuracy ≈73%. They reported a network density of ≃10⁹ junctions cm^−2, and a network size of ≃4 cm². It is difficult to compare simulation results from the current study using 300 nanowires to those obtained with the densities of Demis et al, however the simulation study by Sillin et al [36] provide a better comparison. They did not report accuracy, but reported the lowest mean square error (MSE) of ≈0.12, using a similar density (to Demis et al), however they only simulated 50–400 total connections in the network. Results from the current study show higher square-wave NLT performance of up to 85% (see figure 3) with a significantly reduced number of wires. The improved performance in square-wave NLT likely arises from the topology of our simulated NWNs. Both Demis et al and Sillin et al grew nanowires around grid-shaped copper posts, while our simulation models self-assembled networks. It is possible that the grid-like structure of the copper posts also affects the topology of NWNs.

The MC task was originally proposed specifically for echo state networks (ESNs) [40] and has been tested on networks with different structures [57–60]. Table 1 lists the maximum MC scores attained with different ESNs, compared to MC scores for NWNs with similar properties. The best performance, with a maximum MC of 120, is achieved for an NWN with 300 nodes and average degree of 97. When the average degree is reduced to 6, maximum MC drops to 60, which is still substantially higher than that achieved for a 500-node ESN with the same density. The best performing ESN in this list is for a network prepared in an edge-of-chaos state. While we did not pre-initialise NWNs in this study, we have elsewhere shown that NWNs can be prepared in an edge-of-chaos or critical-like state to improve performance in RC tasks [47, 48]. In those studies, we found a maximum MC of around 12 for 100 nanowires and 100 for 300 nanowires [48], and NLT accuracy ≈0.85 with 100 nanowires [47].

Modular NWNs with 150 nodes in each module achieve poor MC performance when modules are fully separated, however once even a small amount of overlap is allowed (i.e. segregated networks), the NWNs outperform even the best ESN (with max MC of around 90). When compared to modular ESNs [60], this is a large improvement in performance. This difference in performance may be attributed to the internal memory property of memristive junctions, which allows NWNs to outperform ESNs, which typically use mathematical sigmoid functions devoid of memory [61]. It is important to note that some ESNs implement integrator neurons, which provide memory at the neuron level. However, to our knowledge, such implementations in the literature have yet to test either MC or NLT tasks.

An advantage of modularity in complex networks is allocation of network resources [49, 50], which allows segregated networks to vastly outperform random (integrated) networks in multitasking [11]. Such effortless multitasking becomes troublesome when network resources are shared (e.g. when performing two complex logical tasks) [10]. If too many network resources are shared, as in the case of highly integrated networks, multitasking becomes less efficient due to interference or crosstalk [10, 11]. We observe this modular advantage here, with segregated NWNs outperforming random memristive (integrated) networks when multitasking, but not on independent tasks.

The NLT task performs NLT of the input signal by the network. This transformation is achieved via a nonlinear activation function at the memristive junctions once a conductive filament bridge forms [47]. Our results show that when enough junctions are activated to form a signal transduction path from source to drain, the network performs optimally on the task. This state produces an optimal number of linearly separable outputs in the feature space. In a physical NWN, this corresponds to significant inhomogeneities in the voltage distribution, with just enough junctions activated to nonlinearly transform the signal. This is qualitatively consistent with the network being in a critical-like dynamical state [38, 47]. Conversely, at higher densities or voltages, too many junctions are activated and the outputs are not sufficiently linearly separable (i.e. reduced feature space). In the physical network, this corresponds to a homogeneous voltage distribution. This results in failure of the network to perform NLT, which we observe from avg. degree of 170, or at voltages higher than 5 V (for avg. degrees greater than 20).

In contrast to the NLT task, the MC task does not rely on the internal resistive memory of individual junctions, but instead relies on the fading memory of the recurrent network connections [40]. Thus, each task tests the network's ability to recall information from the input signal on different timescales, with reservoir-level fading memory depending on faster timescales than junction-level resistive memory. As such, we see higher MC performance at much higher densities and voltages, which supports findings in our previous work [48]. This also explains NWNs' ability to outperform typical ESN implementations with sigmoidal neurons that lack memristive memory properties [27].

Our results show that some level of crosstalk is beneficial for multitasking in NWNs. This supports the results of other studies on neural network architectures showing effortless multitasking when network resources are not overly shared, and reduced multitasking capacity when too many similar resources are being shared [10, 11]. The MC and NLT tasks applied here operate at different time scales, and rely on different network resources. When applied simultaneously to a modular network, the NLT acts as a modulating signal for the MC, boosting performance. This increased performance is similar to findings from implementations of MTL in RC. Such implementations involve parallel arrays of reservoir networks [62], which show improved performance on the MC task over single reservoir network implementations [63, 64]. However, an important feature of our NWN is the additional resistive memory of individual edge–junctions, which enables the network to perform temporal signal processing tasks involving vastly different timescales.

Our results support the idea introduced by Musslick et al [10] and Petri et al [11] that interference reduces multitasking performance when too many resources are shared (i.e. overlapping modules). We show that interference need not be detrimental and may in fact be advantageous if resources are only marginally shared between modules via sparse adaptive connections (i.e. segregated modules).

Results presented here extend our previous findings [30] that NWNs are more modular than random networks, C. elegans and crossbar arrays, by showing that network structure impacts function. In particular, our results demonstrate the advantage of these modular structures for multitasking, as suggested in brain network studies [49, 50, 54, 55]. These results motivate future implementation of multiple tasks in highly modular networks, in an effort to implement bio-inspired multitasking in other neuromorphic systems. Implementing modular structures in hardware NWNs may be advantageous for performing multiple simultaneous tasks while maintaining low power requirements.

4.1.1. Network comparison

We compared four networks (C. elegans, sparse crossbar, NWNs and random networks), each representing a different type of real-world network. For each network type except for C. elegans we constructed ten unique networks, each sharing the same parameters but with different random seeds.

As in our previous study [30], the C. elegans structural connectivity matrix (277 neurons and 2105 synaptic connections) was adapted from Achacoso and Yamamoto [65], and electron microscope reconstructions by White and colleagues [66].

The sparse crossbar arrays were constructed using the bipartite module from the NetworkX algorithms package [67], keeping the number of nodes and edges as close as possible to the C. elegans structure. This array represents a sparse implementation of cross-bar architectures.

The sample NWN networks were constructed as described in our previous study [30] and was selected from a range of varying networks as they had the most similar number of nodes and edges as the C. elegans.

Edges were modelled as threshold-driven bipolar memristive switches, as described briefly in equation (1) and in more detail elsewhere [37–39, 47, 68, 69]. At each junction, we model electrochemical metallisation via a conductive filament parameter Λ(t). The filament parameter is restricted to the interval −Λ_max ⩽ Λ(t) ⩽ Λ_max and dynamically evolves according to equation (1):

$\begin{equation}\begin{aligned}\frac{\mathrm{d}{\Lambda}}{\mathrm{d}t}& =\begin{cases}(\vert {V}_{\mathrm{j}\mathrm{n}}(t)\vert -{V}_{\mathrm{s}\mathrm{e}\mathrm{t}})\enspace \mathrm{sign}({V}_{\mathrm{j}\mathrm{n}}(t))\quad & {V}_{\mathrm{j}\mathrm{n}}(t)\vert { >}{V}_{\mathrm{s}\mathrm{e}\mathrm{t}}\\ 0\quad & {V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}}{< }\vert V(t)\vert {< }{V}_{\mathrm{s}\mathrm{e}\mathrm{t}}\\ b\enspace (\vert {V}_{\mathrm{j}\mathrm{n}}(t)\vert -{V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}})\enspace \mathrm{sign}({\Lambda}(t))\quad & {V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}}{ >}\vert {V}_{\mathrm{j}\mathrm{n}}(t)\vert \\ 0\quad & \vert {\Lambda}\vert {\geqslant}{{\Lambda}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\end{cases}\end{aligned}.\end{equation} \tag{ 1 }$

Parameters were chosen such that network activation time was comparable to hardware networks. Values used are V_set = 0.01 V, V_reset = 0.001, and Λ_max = 0.15 Vs. b = 10 is a constant defining the decay of the filament. V_jn represents voltage across each junction at time, t.

All junctions are initially in a high resistance 'R_off' state (Λ = 0 Vs). For each junction in the network, resistance switches to 'R_on' state when Λ ⩾ Λ_crit, where Λ_crit = 0.10 Vs is a set threshold. The ratio of these resistance states is R_off/R_on = 10³, with ${R}_{\text{on}}={G}_{0}^{-1}$, and G₀ = (13 kΩ)^−1 is the conductance quanta [48]. Junction electron tunnelling conductance was also modelled as explained in Hochstetter et al [47]

Random networks were constructed using double-edge-swaps using the NetworkX package. Two pairs of two nodes, each with an edge between them, were randomly selected and the edges were randomly swapped between the nodes. This process was performed 50 000 times to ensure that almost all the nodes and edges were rewired. Any self-loops resulting from this process were removed. This process also maintains the degree distribution of the original NWN networks. Consequently the random networks had the same number of nodes, edges and degree distribution as the NWN networks, but with a vastly changed structure (approximately random). For comparison, the average degree of each of the networks was constrained to around 13, the density of the biological C. elegans network.

4.1.2. Density

4.1.3. Modularity

4.1.4. Multitasking

All simulations were conducted on Python v3.7.3 and Matlab v2020a. The networks were used as reservoirs in a RC setting using two RC benchmark tasks: NLT [36] and MC [40]. These tasks leverage different aspects of a reservoir's memory properties: NLT uses a relatively slowly varying continuous input signal that preferentially selects for the internal resistive memory of individual junctions, whereas MC uses noisy, randomly selected inputs that preferentially select for the recurrent network's fading memory [27]. We previously implemented these tasks with NWNs in different contexts [37–39, 48]. The RC tasks were implemented by defining one input (source) node, one grounded (drain) node, and all other network nodes as output nodes. Using the voltages at the output nodes as a readout, we trained a linear regression model using only the readout to fit the respective target signal.

For multitask training using modular networks, we treated each module as a separate network, with one source–drain pair in each module and the rest of the module nodes used as output nodes. This meant that a modular 300-NWN would have two sources, two drains, and around 150 nodes in each module used as read outs for separate tasks (e.g. module 1–150 nodes for MC task, module 2–150 nodes for NLT task)

4.2.1. Nonlinear Transformation

4.2.2. Memory capacity

To measure the structural connectivity of each network, the BCT [70] and NetworkX [67] packages were used. To calculate modularity, we used the community Louvain package of the BCT, based on the Louvain community detection method [71]. The optimal community structure of NWNs was determined by performing a resolution sweep on three sample NWN networks, from which we found that γ = 1.1 best captures the modules in our networks. The exception to this was for the two highly segregated modules used for multitasking, for which we used γ = 0.1.