Stable learning of functional maps in self-organizing spiking neural networks with continuous synaptic plasticity

Abstract

This study describes a spiking model that self-organizes for stable formation and maintenance of orientation and ocular dominance maps in the visual cortex (V1). This self-organization process simulates three development phases: an early experience-independent phase, a late experience-independent phase and a subsequent refinement phase during which experience acts to shape the map properties. The ocular dominance maps that emerge accommodate the two sets of monocular inputs that arise from the lateral geniculate nucleus (LGN) to layer 4 of V1. The orientation selectivity maps that emerge feature well-developed iso-orientation domains and fractures. During the last two phases of development the orientation preferences at some locations appear to rotate continuously through ±180° along circular paths and referred to as pinwheel-like patterns but without any corresponding point discontinuities in the orientation gradient maps. The formation of these functional maps is driven by balanced excitatory and inhibitory currents that are established via synaptic plasticity based on spike timing for both excitatory and inhibitory synapses. The stability and maintenance of the formed maps with continuous synaptic plasticity is enabled by homeostasis caused by inhibitory plasticity. However, a prolonged exposure to repeated stimuli does alter the formed maps over time due to plasticity. The results from this study suggest that continuous synaptic plasticity in both excitatory neurons and interneurons could play a critical role in the formation, stability, and maintenance of functional maps in the cortex.

Keywords: STDP; functional maps; learning; ocular dominance; orientation selectivity; spiking networks; stability.

Figure 1

The complete network model (A) with thalamocortical circuit where thalamic afferents from the LGN activate the principal cells of layer 4 of V1 via geniculocortical synapses. The layer 4 and LGN are both modeled as an E-I network. There are two inhibitory populations in layer 4: the feedback inhibitory population I₁, which does not receive any inputs from LGN but only from the E neurons of layer 4 and the feedforward inhibitory population I₂, which does. The LGN receives spikes from retinal ganglion cells (RGC). The LGN and layer 4 neurons in the model are separated by the dashed line in the figure. This complete model is used for simulating the experience-dependent phase of development. For each network layer, 60% of randomly chosen neurons are injected with background noise in form of currents (I_inj) for 30 ms. A new of set of 60% randomly chosen neurons at all layers are selected again after that and are injected with background noise. This process is repeated throughout all three phases of development. For simulating early experience-independent phase (Phase 1) the complete network is purely driven by the background noise at both LGN and layer 4. For simulating late experience-independent phase (Phase 2) the LGN is activated by spikes due to retinal waves from RGC.

Figure 2

The synaptic connectivity density distribution for the network. (A) From any E-neuron to any other neuron (E or I) in shown in green and from any I-neuron to any other neuron (E or I) is shown in red. (B) There are four types of synapses depending on the pre- and post-synaptic neuron: E → E, E → I, I → E, and I → I. It should be noted that the network has periodic boundary conditions such that topmost and bottommost neurons are regarded as neighbors, as are the leftmost and rightmost columns within each layer. The neighborhood around each E or I neuron is shown as a dotted circle. (C) The LGN network is also an E-I network as shown here with mutually inhibiting connections between neurons that receive inputs from the RGCs (not shown) from the left and right eye. Each LGN neuron from both eyes project to a neuron and its neighborhood in layer 4. For convenience, only one such projection is shown here. The LGN network (2 × 48 × 48) is smaller than the layer 4 network (128 × 128). In addition, the LGN inputs from the left and right eye populations project to the I₂ population in layer 4 which consists of 48 × 48 inhibitory neurons as well. (D) The synaptic connections for a set of 10 × 10 E neurons in layer 4 are shown here. The red square shows a single E neuron with a 19 × 19 neighborhood. The white pixels within each such square indicate synaptic connections with maximum synaptic strength while black pixels indicate synaptic connections with zero synaptic strength. A closer look at the 19 × 19 neighborhood for one of E-neuron shows the initial strengths of synapses from the E-neuron in the center to its neighboring E neurons. These synaptic strengths are randomly distributed.

Figure 3

(A) The E-STDP function modulates the excitatory synaptic conductance w based on the timing difference (t^pre_i − t^post_j) between the action potentials of pre-synaptic neuron i and post-synaptic neuron j. The control parameters (A⁺, A⁻, τ⁺, and τ⁻) can be used to modify the amount of potentiation and depression (see “Materials and Methods”). (B) The I-STDP function modulates the inhibitory synaptic conductance z based on the timing difference (t^pre_i − t^post_j) between the action potentials of pre-synaptic neuron i and post-synaptic neuron j (see “Materials and Methods”). If the timing difference is > λ, then the synapses become less inhibitory and the change itself is of a smaller magnitude while it is the opposite case when the timing difference is <λ.

Figure 4

Model architecture and simulations of first phase of recurrent cortical map (RCM) formation for three different cases are illustrated here. The patterns of the RCM are aligned systematically with the Caretsian grid on which the population is laid out but the emergence of smooth organization is due to synaptic plasticity that changes the connectivity of neighbors to become similar to each other. (A) When STDP in both excitatory and inhibitory synapses are active, RCMs are formed with a clearly developing structure. The RCM is created using the E → E synaptic conductance values (“Materials and Methods”). The average firing rate of the E neurons of layer 4 is ~10 Hz. Low firing rates indicate a good balance between excitation and inhibition. (B) When there is no inhibition, RCMs fail to emerge and the average firing rate of the network is ~100 Hz. (C) In the case of fixed inhibition in the network, RCMs do emerge for some but not for all fixed inhibition settings. The average firing rate of the network for one such example setting of fixed inhibition for which no RCM emerges is ~75 Hz.

Figure 5

Evolution of RCMs during Phase 1 at three different stages of development. (A) After 0.1 million steps of simulation the RCM is well formed but not completely stable. The second stage after 0.5 million steps of simulation shows the emergence of structure in local synaptic connectivity between the E neurons in layer 4 (see “Materials and Methods”). The RCMs are stable after 1 million steps. (B) The synaptic conductances within the pin wheel for a 40 × 40 neighborhood (black square) is shown where there is a clear lack of appearance of structure after the first stage. After a 1 million steps of simulation, the RCM is well formed and the synaptic conductances show well developed structure except at the pin wheels. (C) The histogram of the synaptic conductances in the E → E synapses of layer 4 shows that the initial histogram of synaptic conductance values is bimodal with most of the synapses fixed at a value of 0.12. As the RCM evolves, this bimodal distribution changes due to competition between E → E synapses that is caused by STDP. This competition creates a sparse network with a majority of the synapses becoming zero while a fewer set of E → E synapses are fully potentiated to 1.0.

Figure 6

Various aspects of formed OSM at the end of Phase 1 including orientation gradients, orientation selectivity, and orientation preferences of neurons are shown here. (A) The oriented bar stimuli are provided as input to the LGN neurons and the raw firing rates of the E neurons in layer 4 are measured (“Materials and Methods”). (B) The smoothed responses (see “Materials and Methods”) of orientation preferences is computed and plotted in color. The oriented bars on the right provide the cyclic color code ranging from 0° to 180°. (C) Orientation selectivity of each E neuron is indicated using a grayscale map. The brighter colors indicate high selectivity where those E neurons in layer 4 respond sharply to a very narrow range of orientations and vice versa. The color scale shows the magnitudes of responses in a relative fashion. For example, neurons in the neighborhood of neuron at (110, 100) show strong selectivity (score of 110) to only 120° but not to other orientations while a neuron at (80, 50) shows a weak selectivity (score of 8) to all orientations including its preferred orientation of 150°. (D) The absolute magnitude of the orientation gradients at each E neuron is shown here. Here lighter values indicate high gradients (closer to 90°) while black indicates there neighborhoods have similar orientation preferences and thus no gradient at all. Orientation selectivity and orientation gradients are linked such that regions of high selectivity typically have low gradients while regions of low selectivity have high gradients in a manner qualitatively consistent with the data from the Blasdel paper. The discontinuous changes either occur alone (singularities), or they group together along lines (fractures). While there are many lines of fracture in this phase, there are no singularities, linear zones or pinwheels.

Figure 7

Formation of ODMs during Phase 1. (A) The methodology of constructing the ODMs is outlined here (see “Materials and Methods”). The synaptic conductance changes induced due to STDP at the geniculocortical synapses from the LGN neurons corresponding to each eye are tracked over time. If the E neuron in layer 4 has a stronger set of afferents from the LGN from the left eye compared to the right eye, then the E neuron is labeled as “left” and color coded as green in the ODM. The exact opposite scenario results in the E neuron labeled as “right” and color coded as red in the ODM. If there is a tie (as in the beginning), then the E neuron is labeled as “binocular” and color coded as blue in the ODM. (B) The sum of the synaptic conductances from the LGN neurons corresponding to the left eye is compared against the sum of the synaptic conductances from the LGN neurons corresponding to the right eye at each E neuron. Each of these neurons are labeled as “left” or “eye” as described above. Then the mean and standard deviation of the difference between left and right eye afferent synapses for the neurons (nleft) labeled “left” is computed and plotted in a semilog format with the ordinate plotted in log scale while the abscissa data is plotted in regular scale. It can be seen that the mean value of the difference increases slowly. However, the standard deviation of the difference is higher than the mean implying that many of the E neurons have a very small difference in synaptic conductances while a few have a much larger difference. Similar behavior was observed for the right eye as well. This measure shows that the ODMs are not really stable during the Phase 1 in our model. (C) Early ODM appears to have several binocular E neurons since all the geniculocortical synapses are initialized with the same synaptic strength and there have not been sufficient inputs to alter the synaptic strengths via STDP. (D) At the end of Phase 1, the E neurons in the ODM shows eye selectivity with an even split of neurons becoming selective to one of the two eyes. There are no more binocular neurons. The ODM, however, appears fragmented with no large contiguous areas of neurons showing preference to one eye and not the other. Instead it has lots of small contiguous areas of various sizes. This is due to random stimulation of the LGN neurons with background activity with no temporal or spatial contiguity during Phase 1. The STD and mean data in (B) also provides further support to this basic phenomenon during this phase. However, this improves during the second and third phases of development as shown in Figures 10, 12 when there is more structure in the input data.

Figure 8

The retinal wave model. (A) Shows 128 × 128 RGCs from one retina. After 1 ms of spontaneous firing of RGCs, there are 10 sites that are randomly initiated to generate spikes. These sites create further activity among neighboring RGCs to propagate a wave of spikes as shown for t = 6 ms and t = 10 ms. Each such spontaneously initiated wave activity is terminated at the end of 10 ms. The 128 × 128 retinal wave image is donw-sampled to a 48 × 48 image and provided as input to the LGN. (B) The superposition of all spike activity after 300 retinal waves shows a good spatial distribution of spike activity covering all parts of the retina. (C) The late experience-independent phase in our model lasts for around 200,000 retinal waves when all the RGCs in the retina are activated at least once and the spiking activity of RGCs resembles a Gaussian random field-like distribution.

Figure 9

The formation of RCMs, OSMs, and ODMs under normal development and with lesion in the LGN → E pathway is shown here. The RCMs are the most stable since they are fully dependent on local recurrent connections within layer 4. The OSMs are altered more since their formation is dependent on both local recurrent connections as well as feedforward connections from LGN to layer 4. The ODMs are the most affected by lesions since they are primarily affected by LGN to layer 4 connections. (A) The column shows the RCM, OSM, and ODM after 4 million steps. The eye selectivity of the E neurons in layer 4 appears to be evenly balanced between the two eyes. The ODM appears to have more contiguous patches of neurons that respond to only one of the two eyes. The OSM is developed with more patches of iso-orientation domains. (B) The development of ODMs is affected with more E neurons tuned to the left eye compared to the right eye when 25% of RGCs in the right eye are prevented from stimulating the right LGN neurons. The OSMs also show a reorganization of the map albeit without any dramatic changes. The RCMs are still stable. (C) The development of ODMs is more severely affected compared to (B) when 50% of RGCs in the right eye are prevented from stimulating the right LGN neurons. The OSMs still show iso-orientation domains but there is a re-organization of the patches relative to the 25% lesion case. (D) The development of ODMs is even more severely affected compared to (C) when 75% of RGCs in the right eye are prevented from stimulating the right LGN neurons. The RCMs on the other hand do not appear to be that dramatically affected. The OSMs still show some differentiation in its orientation preferences albeit undergoing more re-organization compared to the 50% lesion case.

Figure 10

ODM and OSM formation during Phase 2. (A) The ODM appears to have large more contiguous patches of eye selectivity compared to the output after Phase 1. (B) This maturity in ODM formation is caused by more stability in the divergence between the mean and standard deviation calculations in the semilog plots shown here. This divergence is very clear unlike in Figure 7. The mean and standard deviation also seem to stabilize in the later parts of Phase 2. (C) The absolute magnitude of the orientation gradients at each E neuron shows singularities and fractures. Here white indicates high gradient values while black indicates no gradient at all. The gradients image show several fractures in the data. The average fraction of the total synaptic drive at each E neuron selective to a given eye was also calculated. For example, for the “left” E neurons, (∑_i w^left − ∑_i w^right)/(∑_i w^left + ∑_i w^right), was ~71%. Similarly, the fraction was ~70% for the “right” E neurons at the end of Phase 2. (D) Orientation selectivity shows brighter colors that indicate high selectivity with those E neurons in layer 4 respond to a very narrow range of orientations and vice versa. The color scale shows the magnitudes of responses in a relative fashion. For example, neurons in the neighborhood of neuron at (85, 80) show strong selectivity (score of 270) to only 30° but not to other orientations while a neuron at (60, 60) shows a weak selectivity (score of 3) to all orientations including its preferred orientation of 60°. (E) The smoothed orientation preference map shows iso-orientation domains and three weakly formed pinwheels marked by the three black circles. We call these weakly formed pin wheels since they are not corroborated by singularities in the orientation gradient maps. This is because the smoothing operation on the Cartesian images removes the spurious edges created by noisy neuron responses while also removing any trace of the singularities as well. However, close inspection shows that there are three locations marked with black circles where the orientation preferences rotate continuously through ±180° along circular paths. We refer to these patterns as a pinwheel-like pattern. It should be noted that there are no clear appearance of point discontinuities in the orientation gradient maps to corroborate the pin-wheel centers within these pinwheel-like-patterns that clearly appear in animal data.

Figure 11

Natural stimuli during Phase 3. (A) The input image from the Caltech database is down sampled from a 320 × 200 image to a 128 × 128 image and pixels from within a 48 × 48 window from the center of the sub-sampled image (red box) are used to create the stereo pair. The resulting image for the right eye is obtained by shifting the fovea to the right and applying a scaling factor of 1.025 on the left image. The resulting stereo pair is finally processed to generate ON and OFF images for each eye (see “Materials and Methods”). (B) A second example of an image from the Caltech 101 database using the same scaling and pixel shift as in (A) is used for extracting the down sampled ON and OFF images. The appearance of the pixels in the ON and OFF images for both examples has similar statistics in terms of the contrast and oriented edges.

Figure 12

Functional maps summary and orientation selectivity of neurons in OSMs during Phase 3. (A) Comparison of functional map development during all three phases shows the progressive refinement of the ODM, RCM, and OSM. The ODM is stable and shows well-defined and contiguous patches of eye selectivity. (B) The mean and standard deviation parameters as described in Figures 7, 10 show more divergence and stability compared to Phase 2 The average fraction of the total synaptic drive for the “left” E neurons was ~123% and ~124% for the “right” E neurons at the end of Phase 3. (C) The resulting orientation gradient maps show well-defined fractures but no singularities. (D) The orientation selectivity has similar characteristic to that of Phase 2 except that the peak magnitude of selectivity showing more sharpness (i.e., higher magnitude). (E) The orientation preference maps show clearly defined iso-orientations and four weakly formed pinwheels marked by four black circles.

Figure 13

The average synaptic current difference between excitation and inhibition is plotted as a function of developmental time. (A) Experience-independent OSM development (Phase 1). (B) Experience-independent ODM development (Phase 2). (C) Experience-dependent refinement of both OSM and ODM (Phase 3). In all the plots, the instantaneous current difference at each time step is shown in blue while the average current difference is shown in red. The plots show that the stability of the functional maps are correlated closely to the fact that the average current differences become progressively smaller as the maps develop. This is enabled by inhibitory plasticity and helps in preventing any rapid changes in synaptic conductances.

Figure 14

The change in synaptic conductance distributions during the course of learning. (A) The E → E change shows a gradual decrease in the change with the lowest values in the third phase indicating stability in the formed OSM and ODMs. (B) The E → I change shows a similar trend as in (A). (C) The I → E change also shows gradual stabilization as a function of development. (D) The I → I change achieve stability very early and remain very stable after indicating that the synaptic conductances between inhibitory interneurons are stabilized rapidly compared to the other three types of synapses. In all the plots, the instantaneous change in synaptic conductance distributions at each time step is shown in blue while the average change is shown in red.

Figure 15

ODM and OSM change in response to new stimuli. (A) Stability of the development process was tested using a sequence of test patterns shown here as stereo pairs with ON and OFF images for each pair. The test patterns consist of mixtures of horizontal and vertical lines that combine to form flag-like patterns. These flag-like patterns are presented for total duration of T_F seconds. The duration of each flag-like pattern was between 10 and 100 ms. (B) OSM, RCM, and ODM maps after presenting the flag patterns for a duration of T_F = 10 s after Phase 3 shows that the maps are stable (compared to Figure 12) despite constant variations in the duration of presentation of each flag-like patterns. (C) OSM, RCM, and ODM maps after presenting the flag patterns continuously for a duration of T_F = 5000 s after Phase 3 shows that the ODM and OSM change while RCM does not change much at all (compared to panel B). The change in orientation selectivity in the early and late stage of input patterns shows a noticeable change in the OSMs. The number of neurons with responses close to 0° or 180° (red and magenta) and 90° (cyan and green) have gone up compared to the plot in panel (B) indicating that the change reflects the dominance of vertical and horizontal bars in the stimuli.