There are measures that can quantify the spatial differences or dissimilarity between the resulting spatial organization of synaptic efficacy from two respective groups for various levels of competition. One such measure is called the spatial dissimilarity index (SDI) (Duncan and Duncan, 1955; Traulsen and Claussen, 2004). This index measures of the dissimilarity or equivalently the overlap between two spatial patterns, where segregated or similar patterns give a value close to unity or zero, respectively. The SDI is formulated as:

where WjA,B are the total synaptic efficacies at dendritic position j contributed by groups A and B, respectively, and WtotA,B are the total synaptic efficacies summed over all dendritic sites for each of these two groups.

To assess the spatial differences between multiple (more than two) spatial patterns, the above described index is replaced by one based upon mutual information. We have previously used the multigroup mutual information index (mMHI) (Iannella et al., 2010), defined as

where subscripts j denotes dendritic location and m indexes the particular afferent group where:

To quantify the degree of spatial segregation between multiple spatial patterns.

A biophysically detailed compartmental model of a reconstructed layer 2/3 pyramidal neuron receiving randomly timed excitatory and inhibitory synaptic inputs across the dendrite, was simulated using the NEURON simulation package (Hines and Carnevale, 2001). The model consisted of 119 sections with 263 segments, including a simplified myelinated axon, similar to those used in previous studies (Mainen et al., 1995; Iannella and Tanaka, 2006), consisting of a hillock, initial segment, five nodes and five myelin internodes, respectively. The parameters and channel types used in the simplified axon were the same as those used in Iannella and Tanaka (2006). A variety of synaptic receptors, voltage and calcium dependent ion channels known to exist in real layer 2/3 pyramidal cells were incorporated into the model. These receptors included the α-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor, the calcium permeable N-methyl-D-aspartate (NMDA) receptor, the ionotropic and G-protein coupled gamma-aminobutyric acid receptors (GABA_A and GABA_B), respectively. The included ion channel currents were a passive leak I_leak, the fast sodium I_Na, the delayed rectifier potassium I_Kdr, the hyperpolarization activated potassium I_h, the transient A-type potassium I_A, a muscarinic potassium I_M, the T-type calcium I_T, (high voltage activated) L-type calcium I_HVA, the calcium dependent potassium C-type I_C, the medium after hyperpolarization (AHP) I_mAHP; and the slow AHP I_sAHP. These channels were distributed throughout the dendrites, soma, and axon with densities according to published experimental data from the rat. In case when data was absent, both distributions and parameter values similar to those used in previous studies were used. Finally, passive properties used in our Layer 2/3 model neuron, were similar or adopted from previous investigations (Mainen et al., 1995; Iannella et al., 2004, 2010): the dendritic membrane capacitance was C_m = 0.9 μF/cm², the resting potential was −80 mV, and the internal resistivity R_a was 200 Ωm.

A full description of the ion channels, their corresponding currents and distributions used in the simulations were similar or identical to those used in previous modeling studies (Rhodes and Gray, 1994; Mainen et al., 1995; Rhodes and Llinás, 2001; Traub et al., 2003; Iannella et al., 2004, 2010; Iannella and Tanaka, 2006). These descriptions are detailed in the Supplementary Materials.

Stimulation to the Layer 2/3 pyramidal cell was provided by a single inhibitory group consisting of 250 afferent fibers and either two of four groups of equally sized groups of correlated excitatory fibers. Here, when two groups were used there were 500 afferent fibers per excitatory group, while for stimulation originating from four groups there were 250 afferents per group. Furthermore, there were no or little correlation between the activity carried by any single afferent fiber from one group and those from any other group. Put simply, the activity within any single group was correlated, but the activity of afferents between different groups were not correlated with each other. Finally, the activity carried by excitatory and inhibitory fibers are also uncorrelated. These excitatory groups will be labeled alphabetically, i.e. for two groups they will be referred to as groups A and B, while for four, they will referred to as groups A, B, C, and D. Whether inhibitory or excitatory, each afferent fiber forms five synaptic contacts at randomly chosen locations throughout the dendrite of the model, as suggested by current anatomical data (Thomson et al., 1994, 2002; Markram et al., 1997a; Feldmeyer et al., 2002).

Each simulation began by allowing each excitatory afferent fiber to connect to five randomly selected positions across the dendrite. Similarly, each inhibitory afferent also made five synapses at locations randomly selected throughout the initial segment, hillock, soma, and dendrite. All synapses were activated at random times. Inhibitory fiber activity was modeled via a temporally homogeneous Poisson processes with a mean frequency of 10 Hz. Excitatory afferent activity was modeled as correlated Poisson processes where the activity of a particular group contains higher order statistics (correlations) (Kuhn et al., 2003). These correlations are mediated by coincident activity involving distinct subsets of fibers that only belong to a single group of afferents, while there is no correlation between the activity of any pair of input fibers that belong to different groups, i.e., the cross-correlation between these fibers is zero. The adopted within group correlation coefficient was set to C = 0.05, with a mean firing rate for all excitatory fibers of 40 Hz, accepted where otherwise stated.

The synaptic weights of AMPA conductances w_j(t) ∈ [0, 1] were altered by STDP, (NMDA, GABA_A and GABA_B conductances remained fixed). Learning was implemented using a nonlinear STDP rule (Gutig et al., 2003). For clarity, this rule is described below.

The formation of spatial patterns displaying a clustered organization typically emerge by balancing the requirements of co-operation and competition of some limited resource. In the case of Gütig's nonlinear STDP (see Materials and Methods—STDP learning rules), synapses compete both spatially and temporally to control the timing of somatic and/or dendritic spike generation. This competition is believed to take the form of a spatio-temporal winner-take-all process that ultimately leads to the formation of synaptic efficacy clusters. A key feature of Gütig's nonlinear STDP rule is the presence of the exponent μ. This parameter controls the weight dependence of the rule and can be interpreted as a parameter that controls the degree of competition, since μ = 0 corresponds to the additive STDP and exhibits strong competition (Song et al., 2000; Song and Abbott, 2001); while μ = 1 recovers the multiplicative STDP rule, a rule known to display stable yet weak competition dynamics (van Rossum et al., 2000). For intermediate values of μ the weight dependence is nonlinear and can be interpreted as introducing some intermediate degree of strong and weak competition.

To assess spatial segregation and complementarity in the case of two independent stream of inputs, we used the spatial dissimilarity index (SDI) (Duncan and Duncan, 1955; Traulsen and Claussen, 2004) (see Section Materials and Methods). Note that two spatial patterns are considered to be complementary when they possess a reciprocal relationship in space so that one pattern can be considered the negative image of the other. This index provides a quantitative measure of the difference between two spatial patterns. Two patterns are similar if the SDI is close to zero, while values close to unity indicates that two patterns are spatially segregated.

The SDI varies over time, in a manner that depends on the exponent μ (Figure 1). Note that for each trial, a different random seed was used to initialize the simulation, assuring that both the pattern of afferent connectivity and the activity seen at any location in the dendrite are different between trials. There is an optimum value for μ maximizing the SDI; for which synaptic competition produced the largest degree of spatial complementarity. Above the optimum, high value of μ weakens synaptic competition, leading to an increase in the average size of the efficacy engram contributed by each respective group of excitatory fibers. Such increases in size increases the overlap between the clusters contributed by each respective group of excitatory afferent fibers, to the point where there is little difference between them. More surprisingly, too high competition also leads to cluster fragmentation and SDI decrease.

Another important facet is to determine how altering the balance between potentiation and depression (A₋τ₋/A₊τ₊ ratio), impacts the formation of clustered synaptic efficacy engrams. Previous theoretical studies, using the simple integrate-and-fire model neuron, have shown that in order to avoid unphysiological increase of synaptic weights, this ratio is important for stable yet competitive learning (Song et al., 2000; Song and Abbott, 2001). However, for a spatial model neuron, the impact of such alterations was unknown. This issue was directly addressed here by increasing the maximal amplitude of the STDP learning window's depression component A₋. Again SDI was averaged from ten trials, each initialized using a different random seed.

The imbalance between depression and potentiation in STDP was increased from 1 to 6, for which synaptic depression is overwhelmingly dominant and results essentially in silencing the neuron. Furthermore, this avoided the situation where, irrespective of the afferent group, the total synaptic efficacy contributed by a single group at any location was zero, i.e., WtotA=0 and/or WtotB=0. Beyond these conditions (ratios > 6), numerical singularities prevented SDI calculation.

Figure 2 indicates that increasing the imbalance from 1.05 to 6 resulted in a strong non-monotonous effect on the emergence of synaptic clusters across the dendrite. Firstly, there is a clear maximum SDI value indicating that for that particular value of the ratio, the degree of spatial complementarity between the two patterns is greatest. Secondly, increasing the ratio to a value of 2.25 eventually leads to a minimum SDI value of ≈ 0.45, with no clear segregation of synaptic clusters. Finally, further increases in the balance ratio A₋τ₋/A₊τ₊ to 3.3 (Figure 2B) and to 6 (Figure 2C) leads to increased spatial segregation of synaptic efficacies across the dendrite, contributed by each afferent group (appearing in Figures 2B,C), and gives rise to a surprising recovery in the value of the SDI, where spatially segregated efficacy clusters (appearing to the right of the central plot) are still present in the dendrite. Intuitively, our expectation was that increasing the imbalance by favoring LTD, would lead to degradation of the clustering; surprisingly, the synaptic efficacy clusters were still present but at the cost of decreased spatial complementarity of the original tiling pattern. Here, complementarity is referred as the amount of spatial segregation and overlap between two patterns where one pattern can viewed as the “negative” of the other, and example is provided in Figure 3 for an imbalance ratio of 1.5 where one group dominates regions in dendritic space where the second group does not. Interestingly, a very steep transition separates the SDI maxima and minima, with an inflection point at 2. While the exact position of the inflection point is irrelevant (as it is probably multi-factorial), this reveals a high sensitivity of the model to LTP/LTD balance that may be detected experimentally.

We have observed the formation of synaptic efficacy clusters when there are only two different groups of afferent axons. When more than two streams of inputs are simulated, altering the balance ratio A₋τ₋/A₊τ₊ admitted by the STDP rule also leads to changes in the formation of dendritic mosaic. To quantify clustering in these multi stream conditions, we have previously identified the multi-group Mutual Information Index (mMHI) to be a suitable metric (Iannella et al., 2010; see Materials and Methods section for details).

Figure 4 shows how STDP imbalance impacts the mMHI for three values of μ = 0.03, 0.08, 0.15, when the neuron is being stimulated by four groups of afferent fibers. Note that the same incremental changes to A₋ used in Figure 2 was adopted. The general shape of the relationship between mMHI and balance ratio is comparable to the one quantified by SDI for two groups of axons and is similar for all values of mu. This dependency is thus robust to change of clustering metric and to increase in the number of inputs, suggesting that this emergent properties could be characterized experimentally (see Section Discussion).

We have shown above that both the degree of synaptic competition and LTP/LTD balance ultimately determines the emergence of the dendritic mosaic. Previously, we have shown that the mean firing frequency of afferent inputs also play an role in this emergence (Iannella et al., 2010).

We thus examined the interplay between firing frequency and STDP balance. Figure 5, illustrates how increasing the degree of imbalance affects the mMHI, for a mean input frequency of 10 Hz. Note that in contrast to Figure 4, the mMHI reaches a plateau (≤0.25) at high ratio rather than showing monotonous increase. Equally surprising is the apparent lack of correlation between the low mMHI and the pattern of synaptic efficacy clusters. For a balance ratio of 1.8 (Figure 6, Row B) corresponding to maximal mMHI, we observed intense clustering, with most dendritic compartments being dominated by a single input stream (bright color). Clearly, these inputs have been potentiated through STDP while the inputs from other streams have been depressed.

For lower (A) and higher (C) balance ratio, this pattern is degraded, with a decrease of the contrast between clusters and appearance of overlaps (regions where two or more stream remain strong, thus shown in black). For a balance ratio of 6 (Figure 6C) we even observed subsections of the dendritic tree where efficacy clusters were absent. These dendritic subsections had essentially become regions that do not respond to any group of afferent inputs.

Increasing both the mean input frequencies and the degree of imbalance induce non-trivial changes for the mMHI (Figure 7). Above 20 Hz, a local mMHI minimum replaces the steady decay seen at lower frequency. The value of the mMHI minimum is approximately independent of frequency but frequency determines the balance ratio at which the minimum occurs. The mMHI maximum remains at 1.8 A₋τ₋/A₊τ₊ for all frequencies. Note that already at 40Hz, the upward slope of the mMHI tend toward an asymptotic limit. Indeed no further change were observed at frequencies >40Hz. Thus, the mMHI in the 0 ~ 2.5 A₋τ₋/A₊τ₊ range is similar for all input frequencies and below and in the 2.5 ~ 6 range for input frequencies above 30 Hz. This suggests that the conditions required for the optimal tiling of synaptic clustering are robust for a wide range of parameters. The unexpected appearance of a local minimum at ~2.5 may be linked to the degree of local excitability, the number of post-synaptic events generated by incoming inputs and where they occur in dendrites. The number of generated action potentials clearly follows a nonlinear relationship with the mean frequency of the inputs locally targeting the dendrite.

In the previous section, we saw how the mean input frequencies and the degree of STDP balance affects the emergence of the dendritic mosaic. Here, we analyze the corresponding alterations to local neuronal response at the soma and several representative dendritic locations, as shown in Figures 8A,B. Firing rate at the soma (when subjected to inputs from all four afferent groups) dramatically decreases from 86 Hz to 0.2 Hz as the degree of imbalance (balance ratio) increases from 1.05 through to 1.8 (Figure 8A).

For larger degrees of imbalance (from 1.8 to 6) the neuronal firing rate remains relatively constant around 0.03 Hz. In contrast, the firing rate for the dendritic locations display maximal spikes rates at different values of the balance ratio (Figure 8B) and different level of activity, even when LTD is favored (high STDP imbalance ratio).

In the realistic model presented above, we have measured how STDP balance and mean input frequency influence the emergence of efficacy clusters. Since clustering emerge from local interactions of synaptic inputs, we wondered to what extend dendritic morphology plays in the appearance and stability of the mosaic. Historically, Rall (1964) was the first to show how neural morphology can influence the firing properties of neurons, using simple compartmentalized ball-and-stick models (Rall, 1964). To directly investigate whether the morphology of the dendrites impacts the emergence of the dendritic mosaic, we chose to compare a realistic (reconstructed from tracing) neuron morphology to an extremely simplified simple cable equivalent. This extreme approach allowed us to preserve the active properties of the original cell with only simple transformations while reducing the morphology to an unbranched dendrite. We thus generated the simplified models using two published reduction methods that map complex dendritic morphologies to an unbranched cable structure while maintaining axial resistance and without altering the active properties used in the original model. An alternative approach could be to systematically change the lengths and diameters of dendritic branches, or to gradually merge branches together until significant mosaic alterations are detected. This approach has the double handicap of being both computationally intensive and not amenable to experimental testing. Instead, in a future study, we intend to compare mosaic formation in realistic but different morphologies such as CA1 pyramidal cell, Purkinje cell and spinal cord motoneurons and compare simulation data to physiological data.

The first of these methods relies upon a very simple construction yielding a simplified equivalent cable morphology of the original dendritic tree, consisting of only three identified regions but conserving the electrical properties, length and total surface area of the original cell (Iannella et al., 2004), as shown in Figure 9. The second method is based upon combining branches into an equivalent cylinder where the axial resistance of the original branches are not altered (Destexhe et al., 1998). This requires summing the cross sectional area of each contributing branch and results in an equivalent cylinder with a radius given by the square root of sum of all contributing radii squared, r=∑iri2. The length of the cable is taken to be the average length of contributing branches weighted by their respective radii r_i, in order to take different contributing branch lengths into account, l=∑iliri/∑iri. The resulting total surface area of the simplified model differs from the original, requiring correction of all conductances and the membrane capacitance by a multiplicative factor to maintain input resistance (see Destexhe et al., 1998 for more details). The Destexhe simplified morphology is presented in Figure 9.

Figure 10 presents the mMHI index as a function of STDP imbalance for both of these simplified models. Altering the morphology while keeping the same active properties alters the mMHI (Compare to Figure 2). Notably, for the first method, there is an initial sharp drop followed by a slower monotonic rise in the mMHI index while for the second method, mMHI is a monotonic function of balance. Both mMHI profiles are less complex than profile obtained for the complex model (Figure 2), showing that dendritic morphology plays an important role in the emergence of the dendritc mosaic and its constituent synaptic efficacy clusters.

Increasing the degree of imbalance (by increasing the balance ratio) leads to different quantitative effects on the resulting spatial patterning of synaptic efficacy for each simplified model. For the Iannella reduced model one clearly sees that for an initial balance ratio value of 1.15 there is a preference to form large spatial clusters that occupy distinct portions of the unbranched dendrite, thus indicating that for this balance ratio, STDP implements an underlying spatial winner-take-all process that allow large spatial clusters to emerge. The formation of large spatial clusters, however, is a transient occurrence since a small increase in the balance ratio from 1.15 to 1.33 leads to the emergence of smaller clusters spread throughout the dendrite that can overlap in space with other clusters contributed by other groups.

This is in stark contrast to the formation of large distinct clusters. The appearance of large spatial clusters usually indicates synergy between synapses however the transformation from large to small clusters signifies a sudden loss of spatial extent of co-operativity between synapses. Increasing the balance ratio further leads to small synaptic efficacy clusters that are sparsely distributed but can overlap with other efficacy clusters contributed by other afferent groups. In comparison, the Destexhe simplified model results in only small clusters that are spread out throughout the entire extent of the unbranched dendrite, but as the balance ratio is increased, this gives rise to small efficacy clusters that can spatially overlap with other clusters (contributed by other afferent groups) but are also freely distributed along the extent of the dendrite.

Inspecting the spatial organization of clusters that have emerged in both simplified models reveals some interesting common traits. For both simplified models, increasing the degree of imbalance leads to the formation of small localized synaptic efficacy clusters that are sparsely distributed in dendritic space. Notably, when balance ratio is large, there are spatial regions in the dendrite that are devoid of any input (see Figure 11 these are indicated by black arrowheads), while in other regions there can be two or more localized clusters that overlap, potentially mutually augmenting their inputs (see Figure 11 indicated by red arrows).

Since there is a drastic change of synaptic efficacy, we wondered about the concurrent alterations the neuron's input-output relationship in both the space and time domains. Samples of the membrane potential recorded from three different locations, the soma and two different dendritic locations respectively labeled as “S,” “Pos 3,” and “Pos 5” as indicated in Figure 12 before and after STDP learning for both simplified models. Figure 12 illustrates these membrane potential traces for each model at two different values of the balance ratio A-τ-A+τ+ = 1.15 and A-τ-A+τ+=6.0 denoted by A1 and A4, respectively. Here, when a balance ratio of 1.15 is used, one can observe small qualitative changes to the membrane potential at the above described locations after STDP. Conversely, a large balance ratio value leads to marked qualitative alterations to these membrane potential traces after STDP (A4). Specifically, the membrane potential tends to the resting state, with the occasional occurrence of spikes or burst of spikes, at the three specified locations.

There is renewed interest in investigating how synaptic plasticity applied to passive or active dendritic trees shapes the storage of input features via the formation of dendritic compartmentalization (Rabinowitch and Segev, 2006a,b). For example, Tamosiunaite et al. (2007) have reported a type of winner-take-all competition between dendritic branches. In contrast to our study and that of Legenstein and Maass (2011), they only considered the case where individual input groups were initially spatially segregated and would target single branches (Tamosiunaite et al., 2007). More realistically, Legenstein and Maass (2011) have directly implemented neuronal mechanisms where both functional compartmentalization of input features takes place and dendritic branches compete and behave as individual computational subunits. These authors have shown that when explicitly incorporating two levels of competition, one between dendritic branches and the other at the winning branch between locally situated synapses, correlated synaptic activity is strengthened while the efficacy of other synapses decays with time. This permits the features of single input pattern to be stored locally even in one single branch. This type of self-organization thus allows the storage of multiple input features by a single neuron in different non-overlapping regions of the dendrite.

The functional benefits of spatially organized synaptic inputs are well documented (Rall, 1964; Mel, 1992b). The precise biophysical mechanisms leading to this organization, allowing memory engram to be represented in “synaptic clusters” are thus of prime interest, both from biological and theoretical perspectives. Previous studies of neural networks self-organization into functional maps have shown that non-linear interaction of short-range excitation and longer-range inhibition are fundamental in the emergence of any type of clustered organization (Haken, 1977; Kohonen, 1982; Haken, 1983). This interaction is important since it sets up a winner-take-all mechanism which, under the appropriate conditions, ultimately leads to the development of the functional clusters thus forming the map. The interaction function is spatial in nature and usually balanced in the sense that positive and negative areas of the function are (nearly) equal and thus balance each other out. This type of interaction is also envisioned to occur in dendritic trees.

Our study describes the conditions under which synaptic efficacy clusters may emerge to form a dendritic mosaic. We found that it is jointly determined by multiple factors, including mean input frequency, the degree of synaptic competition, synaptic balance and dendrite morphology. The region in this multidimensional parameter space where both synaptic efficacy clusters and the dendritic mosaic emerge as a result of STDP learning correspond to a physiological range (frequency) or range used by others. Future work may consider investigating the role of synaptic balance when both excitatory and inhibitory synapses undergo plastic change, to elucidate how balance and plastic inhibitory synapses jointly impact the spatial organization of both excitatory and inhibitory synaptic efficacies.

A novel extension may be to investigate other forms of morphological influences. Recent studies have pointed out additional influence of neuronal morphology on synaptic plasticity and on formation of cortical circuits. For example, important differences in the shape and distribution of dendritic spines along neuronal dendrites between pyramidal cells from different cortical areas, layers, and species have been observed (Murayama et al., 1997; Elston, 2003; Bianchi et al., 2013; Elston and Manger, 2014). In primates, for example, both the numbers, density, and distribution of dendritic spines differ for pyramidal neurons in different cortical areas, while in the mouse the spine density seems to be constant (Murayama et al., 1997; Elston, 2003; Ballesteros-Yáñez et al., 2006; Benavides-Piccione et al., 2006; Bianchi et al., 2013; Elston and Manger, 2014). Another aspect are dendritic spines, as these are the loci where the formation and refinement of cortical circuitry through processes of synaptic plasticity and (consequently) synaptic transmission takes place. In basal dendrites, it has been reported that the size of the spine head is proportional to the number of post-synaptic receptors and pre-synaptic docked vesicle, while the length of the spine neck seems to be associated with the calcium compartmentalization (Ballesteros-Yáñez et al., 2006). Taken together, all these point to the need to re-examine the role played by neuronal morphology in brain development, especially the impact of developmental morphological changes, synaptic plasticity, and synaptogenesis has on the formation of cortical circuits including the patterning of convergent afferent inputs to neurons. Although such a study would be highly valuable, care needs to be taken to incorporate biophysically meaningful processes that correctly capture the biochemical processes of activity-dependent synaptic plasticity, neuronal growth, and spine creation and elimination. The results of such a study will be presented in a future publication and is thus beyond the scope of this paper.

Another extension would be to base the STDP plasticity on an explicit biophysical model of calcium dynamics and biochemical signaling cascades involved in learning and memory. One candidate is the plasticity rule first introduced by Graupner and Brunel (2007). Using such a rule would allow to correlate the emergence of synaptic efficacy clusters with the underlying states of the biochemical signaling cascade, generating experimentally testable predictions. Other potential improvements are an explicit modeling of dendritic spines and of the underlying reaction-diffusion processes, along with Graupner's calcium-based plasticity rule (Graupner and Brunel, 2007). We expect that the spatially restricted calcium signals may allow the emergence of synaptic mosaic for more streams of correlated inputs, beyond the two and four groups tested in the current model.

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