Idealized model of the developing visual cortex

Recent experiments in the developing mammalian visual cortex have revealed that gap junctions couple excitatory cells and potentially influence the formation of chemical synapses. Though gap junctions between inhibitory cells are ubiquitous in the adult cortex, and their presence has been shown to promote synchronous network firing, their function among excitatory, pyramidal cells remains poorly understood. During development, pyramidal cells that were derived from the same progenitor cell, called sister cells, are preferentially connected by a gap junction during the first postnatal week, while chemical synapses are still being formed. Additionally, these sister cells tend to share an orientation preference and a chemical synapse in the adult cortex, a property that is diminished when gap junctions are blocked. In this work, we construct an idealized model of the mouse visual cortex during the first two postnatal weeks of development to analyze the response properties of gap-junction-coupled cells and their effect on synaptic plasticity. Further, as an application of this model, we investigate the interplay of gap-junction coupling and synaptic plasticity on the order, or organization, of the resulting cortical map of orientation preference. Author summary Gap junctions, or sites of direct electrical connections between neurons, have a significant presence in the cortex, both during development and in adulthood. Their primary function during either of these periods, however, is still poorly understood. In the adult cortex, gap junctions between local, inhibitory neurons have been shown to promote synchronous firing, a network characteristic thought to be important for learning, attention, and memory. During development, gap junctions between excitatory, pyramidal cells, have been conjectured to play a role in synaptic plasticity and the formation of cortical circuits. In the visual cortex, where neurons exhibit tuned responses to properties of visual input such as orientation and direction, recent experiments show that excitatory cells are coupled by gap junctions during the first postnatal week and are replaced by chemical synapses during the second week. In this work, we explore the possible contribution of gap-junction coupling during development to the formation of chemical synapses both into the visual cortex from the thalamus and within the visual cortex between cortical cells. Specifically, within a mathematical model of the visual cortex during development, we identify the response properties of gap-junction-coupled cells and their influence on the formation of the cortical map of orientation preference.

Gap junctions (GJs), or sites of direct electrical coupling between neurons, are present 2 in the primary visual cortex (V1) at many stages of life, from infant to adulthood. In 3 the adult cortex, gap-junction coupling among local, inhibitory cells has been shown to 4 promote synchrony, which has been hypothesized to be important in many cognitive 5 processes such as learning and memory [1,2]. Though GJs have been measured between 6 excitatory, pyramidal neurons in the adult cortex [3], there are very few experiments and 7 the couplings were found to be very rare; consequently, their function remains unclear. 8 Recent experiments show that pyramidal cells are coupled by GJs during the first 9 postnatal week of development, a time at which chemical synapses are highly plastic 10 and are just beginning to develop, leading to a question about a potential relationship 11 between GJ coupling and the development of V1 neuron response properties. 12 One example of a neuron response property in the primary visual cortex is 13 orientation preference (OP), where neurons preferentially respond to the orientation 14 angle of a visual stimulus. In some higher-level mammals such as monkeys and cats, the 15 visual cortex contains an ordered map of the orientation preference of each neuron, 16 where cells preferring similar angles reside close to one another. In rodents, however, the 17 map of orientation preference appears random and disordered, with little correlation 18 between preferred orientation and location in cortical space. 19 This disordered OP map forms early in development and is dependent upon the 20 synaptic connections from a region of the thalamus called the Lateral Geniculate 21 Nucleus (LGN), which begins forming synaptic connections with the cortical V1 cells 22 shortly before birth [4]. During the first postnatal week, pyramidal cells in V1 are 23 lacking recurrent, or cortical-cortical, synapses; however, sister cells, or pyramidal cells 24 that were derived from the same progenitor cell, are coupled through GJs [5,6]. The 25 strength and coupling probability of these GJs decreases steadily during the first 26 postnatal week such that no couplings are detected in the second postnatal week [7]. 27 During the second postnatal week, GJs between sister cells in V1 disappear and 28 synapses begin to form between all cortical cells. Specifically, glutamatergic synapses 29 form among the pyramidal cells, while GABAergic synapses begin to form among 30 inhibitory, fast-spiking (FS) interneurons. In addition to the GABAergic synapses, FS 31 cells also develop GJ coupling beginning in the second postnatal week and increasing in 32 strength over time [8]. Turning to the map of orientation preference, visual input is not 33 necessary for cortical cells to develop OP [9,16]. Instead, spontaneous activity in the 34 cortex is generated from intra-cortical circuits, as well as input from spontaneous retinal 35 waves [10], and drives synaptic plasticity during the first two postnatal weeks [11]. By 36 the end of the second postnatal week, a weak OP map has already developed and 37 becomes further stabilized by visual input through the newly-opened eyes. 38 Our aim in this work is to better understand how the developmental timeline, 39 including GJ-coupling among sister cells, might affect the formation of a random or 40 disordered OP map. We develop an idealized model as a conceptual realization of a 41 local patch of V1 during the first two postnatal weeks of development. Our model 42 follows the set-up of Ref. [12] and the model timeline of Ref. [11], but with significant 43 adjustments and parameters appropriate for V1. In particular, our model includes spike 44 timing-dependent plasticity (STDP) of the feedforward synapses from LGN to V1 45 during the first postnatal week, as well as plasticity of the cortical-cortical recurrent 46 excitatory synapses within V1 during the second postnatal week. Using this model, we 47 reproduce experimentally-measured properties of GJ-coupled sister cells, such as a form, proposing a mechanism for the "salt-and-pepper" random OP map observed in 53 mice. We also identify mechanisms by which this OP map can become ordered as 54 observed in higher-level mammals, further supporting our proposed mechanism for the 55 development of disordered OP maps. 56 Methods and Models 57 Broad overview of experiments that measure connections 58 between sister cells 59 Gap junctions between excitatory cells in the developing visual cortex of mice have only 60 recently been discovered and their properties measured. The motivation behind this 61 discovery was in uncovering a functional column, such as the orientation hypercolumns 62 in monkey and cat visual cortex, which seem to be lacking in mice and rats. Because of 63 the radial unit hypothesis, a theory positing that the cortex develops as an array of 64 cortical columns due to clonally-related neurons traveling along the same glial fiber 65 (axon of the progenitor cell), experimentalists began investigating the possible functional 66 similarities between sister cells, or cells that stem from the same progenitor cell. Despite 67 the seemingly random lateral (within layer) distribution of OPs in the visual cortex of 68 mice (salt-and-pepper), it has been shown that radially-distributed clonally-related cells 69 show similar stimulus feature selectivity [6], as well as preferential synaptic connectivity 70 with fellow sister cells [5]. Each of these characteristics, measured in the adult cortex, 71 relies on gap-junction coupling between sister cells during the first postnatal week [7,13]. 72 In were labeled by the virus to be sister cells and whose cell bodies were radially-aligned 77 (columnar structure). They specifically measured from clusters of radially-aligned sister 78 cells that were isolated into columns of a tangential width of about 100 µm to be 79 confident that they could distinguish between sister cells from different 80 progenitors.Their results found that no synaptic couplings were detected between any 81 excitatory cells, sister or non-sister, during the first postnatal week (P0-P6). In the 82 second postnatal week, however, sister cells were measured to be coupled with an 83 average probability of 36%, while neighboring (also radially-aligned) non-sister cells 84 were found to be coupled with an average probability of about 6.3% (averaged over P10 85 to P17) [5]. The black lines in Fig 1A show the average percentage of 86 synaptically-coupled sister cells during the second postnatal week.

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A follow-up study conducted by the same group of experimentalists used quadruple 88 whole-cell recordings to show that these radially-aligned sister cells are coupled by GJs 89 during the first postnatal week when chemical synapses are absent [7]. Specifically, they 90 showed that the sister cells are preferentially coupled by GJs during the first postnatal 91 week (28.2% for sister cells compared to 2.6% for non-sister cells, averaged over P1 to GJs are detected between FS cells from P3-P5, with the exception that one functional 102 GABAergic synapse (out of 13 tested pairs) was detected at P5 [8] No recordings were 103 performed before P3. Therefore, we determine that both GJ and synaptic coupling 104 among FS cells are absent during the first postnatal week and grow during the second 105 postnatal week, as shown by the red and green lines in Fig 1A for [14] and that blocking this protein (essentially closing 112 the channel) largely eliminates electrical coupling among sister cells (reducing the 113 average probability over the first week from 26% in wildtype mice to 9.8% in the 114 Cx26-blocked mice) [7]. To test if GJ coupling might be responsible for the preferential 115 synaptic coupling that occurs during the second postnatal week, Yu et al. repeated their 116 earlier experiment of measuring synaptic coupling among sister cells for the case of 117 Cx26-blocked mice as well. They found that the synaptic coupling probability dropped 118 from the typical 30-35% between sister cells in wildtype mice to 8.2% between sister 119 cells in Cx26-blocked mice (averaged over P10-P21) [7], demonstrating that GJ-coupling 120 during the first postnatal week is critical to the correct circuit formation in adults.

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Relatedly, experiments show that excitatory cells that share a similar orientation 122 preference (OP) have an increased likelihood to also be synaptically coupled [15]. Since 123 GJ coupling during the first postnatal week is necessary for preferential synaptic 124 coupling in adults, several experimentalists set out to assess the role that GJ coupling 125 might play in stimulus feature selectivity, such as orientation preference [13,17] than 30 • ), while neighboring non-sister cells exhibit a difference in OP distribution that 130 was not significantly different from the uniform distribution [13]. When a GJ blocker 131 was employed, the effect was destroyed; the distribution of OP difference for sister cells 132 was no longer significantly different from the uniform distribution or the non-sister cell 133 distribution. Figure 1B [5]). Then, sister cells become sparsely intermingled in the mouse visual 140 cortex, with sister cells outnumbered by non-sister cells in a local volume (100-500 µm 141 in diameter) by a factor of six [18,19], a property that seems to be essential for proper 142 synaptic development [20]. While previous experiments concentrated on small groups of 143 radially-aligned sister cells within a radius of about 100-120 µm  In this section, we describe our idealized model of development. Specifically, we use a 162 similar framework as the model in Ref. [12] for the study of spike-timing-dependent 163 plasticity (STDP), but incorporate a more realistic and stable STDP learning rule for 164 the visual cortex adapted from Refs. [22,25] with added inhibitory plasticity as in 165 Ref. [28]. The details are as follows. 166 We consider 1000 feedforward synapses, representing input from LGN to the visual 167 cortex, coupled to our model neuronal network of either 400 or 256 cortical cells. The 168 cortical neurons are randomly assigned to be excitatory with 80% probability or 169 inhibitory with 20% probability. The subthreshold voltage of the ith cortical neuron of 170 type Q = {E, I} is described using the leaky integrate-and-fire equation as follows where τ m = 20 ms, v l = −60 mV, v E = 0 mV, and v I = −80 mV. Once the voltage 172 reaches a threshold of −45 mV, the neuron is said to have spiked, the spike time is response to an action potential in the pre-junctional cell, a 1 mV instantaneous jump in 179 voltage of the post-junctional cell is included, as in previous models [11,21].

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The cortical synaptic conductances are modeled as having instantaneous rise times and exponential decay at each received spike time so that the excitatory and inhibitory conductance traces, respectively, follow the equations respectively, for all cells, whileḡ i EE andḡ i EI are plastic, changing with rules defined in 189 the following subsection.

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The external drive to the cortical network has two components: synaptic input from the LGN and a generic background drive to all cells. This external drive affects the excitatory conductance, g i QE , as follows LGN is plastic, butḡ i back is constant at 0.02. The spike times of the background 191 drive are generated from a Poisson process with rate 0.5 Hz. Each feedforward LGN 192 synapse generates spikes using a Poisson spike train with a firing rate that depends on 193 its own label. Specifically, the firing rate of LGN synapse labeled a in response to a 194 stimulus at input location s is given by as in [12], where R 0 = 5 Hz, R 1 = 20 Hz, and σ = 80. Input to these synapses consists 196 of brief presentations of a uniformly randomly-chosen stimulus index (a in above LGN andḡ i EE , respectively, obeying the minimal triplet rule for the visual 206 cortex [22]. We use the triplet rule rather than the standard pre-post STDP rule that was used in [12] because we wish to reproduce the realistic bi-directional coupling that 208 develops in the visual cortex of mice, a feat that cannot be accomplished with the 209 pair-based STDP rules due to their nature of developing only unidirectional synapses.

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In addition, experiments show that the STDP curves exhibited by pyramidal cells in the 211 visual cortex of mice do not follow the typical slightly-asymmetric shape of potentiation 212 and depression as in [23], but rather potentiation only occurs if the post-synaptic 213 neuron had recently fired a spike of its own [22,24].

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The triplet rule is illustrated in Fig 2A and described as follows. For each pre-and post-synaptic spike, the strength of the synapse from the pre-to post-synaptic cell,ḡ, (dropping the EE subscipt) is updated as follows: where A LT D and A LT P represent the strength of depression and potentiation, respectively. The tracer variables follow the equations where r 1 (t) represents a pre-synaptic tracer, while o 1 (t) and o 2 (t) represent 215 post-synaptic tracers. Note that each neuron carries its own tracer variable, but the i 216 index has been dropped here for clarity. The timescales of these tracer variables are as 217 follows: τ LTP = 16.8 ms, τ LTD = 33.7 ms, and τ trip LTD = 114 ms. To stabilize network 218 activity, we implement a homeostatic mechanism in the form of a rate detector that acts 219 on a fast timescale, known to stabilize the dynamics induced by the minimal triplet rule 220 into recurrent excitatory networks [25]. This homeostatic mechanism works by allowing 221 the amount of depression, A LTD , to change as a function of a moving-average of the 222 post-synaptic firing rate,μ E : where the timescales τ LTP , τ trip LTD , and τ LTD , are those from Eqs (4) - (6), and ρ is the 224 target firing rate, chosen to be 8 Hz to replicate the low firing rate of the mouse visual 225 cortex during early development [27]. The moving average of the firing rate,μ E (t), is 226 found by taking a low-pass filter of its spike train as follows where t k represents the kth spike time that occurred prior to the current time t and 228 τ = 1 s. Note that the synaptic strengthḡ in Eqs (2) and (3)  In addition to the plasticity introduced on the feedforward and recurrent excitatory 234 synapses, we include plasticity on the synapses from inhibitory neurons to excitatory 235 April 22, 2020 7/22 neurons in the cortex [28]. The motivation behind this inhibitory plasticity is that we 236 found it necessary for the inhibition in the network to mediate the excitation for proper 237 development to occur. If the inhibitory synapses were constant at a high value, then the 238 cortical cells would not fire any action potentials. However, if the inhibition was 239 constant at a low value, then as the excitatory recurrent synapses grew, the firing rate 240 of the network would also grow, and the network would become unstable, a common 241 phenomena for plastic recurrent excitatory networks. Therefore, we chose to model 242 inhibitory plasticity as a stabilizing mechanism, as has been done previously in Ref. [28]. 243 A schematic of this inhibitory plasticity can be found in Fig 2A. The synapse from a pre-synaptic inhibitory cell to a post-synaptic excitatory cell updates according to the following ruleḡ where A iSTDP is the learning rate and ρ = 8 Hz is the target firing rate of the excitatory 244 cells [the same as in Eq (7)]. Each cell has a tracer variable where τ iSTDP = 20 ms and x Q (t) → x Q (t) + 1 at each spike time of the cell, similarly to 247 the tracer variables r 1 , o 1 and o 2 in Eqs (4) - (6). Note the interpretation of these 248 plasticity rules: when the spiking of a pre-and post-synaptic inhibitory and excitatory 249 cell, respectively, occurs within a time window of τ iSTDP , either potentiation or 250 depression occurs at each pre-synaptic (inhibitory) spike [as per Eq (8)], while only 251 potentiation occurs at each post-synaptic (excitatory) spike [as per Eq (9)].

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Week 1 Week 3 LGN LGN GJ coupling between sister cells    Fig 2B). Then, once simulation is in the second postnatal week, gap junctions 256 are turned off [by setting g c,E = 0 in Eq (1) input preference (called the OP in this work). We note that the network operates in an 262 asynchronous regime known to accentuate the performance of STDP [26]. synapse formation. Specifically, we demonstrate that a GJ between cortical cells allows 292 those cells to develop a similar OP and a preferential bidirectional synapse, as shown in 293 experiments. In addition, we illustrate that GJ-coupled cells develop an OP at a faster 294 rate than those that are not coupled by a GJ, leading us to hypothesize that the 295 inclusion of GJs during the first postnatal week will enhance spatial disorder in the 296 formation of the OP map.

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To illustrate these effects, we use a 400-neuron idealized cortical network in which 298 20% of the cells are inhibitory and 80% are excitatory. We allow two excitatory cells to 299 be coupled by a GJ with a 50% probability such that about half of the excitatory 300 population is GJ-coupled in pairs (similarly to the small network explored in Ref. [11]). 301 Following the experimental timeline (recall Fig 1A and Fig 2B), we simulate the first bidirectional synapses between GJ-coupled cells is much higher in the model than those 322 observed in real cortex (26% in [7] compared to almost 80% here) since we are directly 323 comparing GJ-coupled cells, while the experiments tested all sister cells (only a fraction 324 of which are coupled by a GJ). 325 We noticed in our simulations that the GJ-coupled cells tend to develop an OP much 326 sooner than the non-GJ-coupled cells. This means that the feedforward synapses from 327 LGN onto the GJ-coupled cells learn much faster than those synapses onto cells that are 328 not GJ-coupled, see Fig 4A for one example cell that was GJ-coupled and one that was 329 not. To see this effect over all GJ-coupled and non-GJ-coupled pairs, for each cell, we where R pref is the firing rate of the neuron at its preferred orientation and R orth is the 348 firing rate of the neuron at the orthogonal orientation (in this work, the orthogonal 349 orientation corresponds to the orientation that is 500 units away from R pref ). An OSI 350 value close to 1 indicates high selectivity and a value close to 0 indicates no selectivity. 351 Figure 4C shows the average of these three properties over all GJ-coupled (blue) and  six [17]. We assume that 256 neurons corresponds to a small enough volume of the 387 cortex that we can consider only six groups of sister cells that are randomly distributed 388 in the space. Within each sister-cell group, each neuron has a 5% probability of being 389 coupled to a sister cell by a GJ. Figure 6A shows a count of the number of cells in each 390 sister group along with the probability of GJ coupling in each group. Note that this 391 coupling percentage is much sparser than the ∼ 28% coupling probability measured 392 experimentally for radially-aligned sister cells [7]. We found that is was necessary to 393 require a sparse GJ-coupling during the first postnatal week for the GJ-coupled cells to 394 exhibit the experimentally-measured properties of OP sharing and preferential synaptic 395 coupling. We explore the effects of larger GJ-coupling percentages in the Supporting 396 information.

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The response properties measured for the pairwise GJ-coupled 400-neuron network 398 remain in this 256-neuron network, including the increased rate of learning for 399 GJ-coupled cells compared to non-GJ-coupled cells, see Fig 6B, and the preference for 400 GJ-coupled cells to share an OP, see Fig 6C. The recurrent synapses between excitatory 401 cells can be all-to-all, but due to the competitive STDP rules, each excitatory cell forms 402 a strong synapse with only about half of the other excitatory cells (the other half decay 403 just as in the feedforward synapses). Figure 6D  We use this network to begin to test our hypothesis that GJ-coupling during the first 408 phase of development leads to disorder in the OP map. First, we show that if the GJs 409 are turned off during the time that LGN synapses are learning (the first phase of 410 development), the distribution of OPs that forms has more order than the one that 411 forms when GJs are present during the first phase of development. Recall that we do 412 not include any spatial effects in this second realization of the model, and we allow any 413 cell in the cortex to form a synapse with any other cell. Thus, to measure order in the 414 OP map, we use the idea from Ref. [12] that when cortical recurrent connections are 415 allowed to form between all cells in the network, all cortical cells develop a similar OP. 416 Figure 7A shows that the OPs in the network without GJ coupling tend to cluster 417 around one value (∼ 375), indicating that the recurrent connections influence the 418 resulting OP of each cell, while the network with GJ coupling during the first phase of 419 learning has a more uniform distribution of OPs. To quantify this, we calculate the 420 distance between each resulting OP distribution and the uniform distribution (sum of 421 the squared difference). Notice that the network without GJ coupling during the first 422 phase of development has a distance of 0.182, while the OP distribution for the network 423 containing GJ coupling has a distance about half of that value at 0.099. This indicates 424 that the inclusion of GJ coupling during the first phase of development results in an OP 425 distribution closer to the uniform distribution, where each OP has equal likelihood of 426 occurring.

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In addition to GJ coupling, the time at which recurrent synapses begin to learn (the 428 start time of the second phase of development) also has an effect on the distribution of 429 OPs. Specifically, the amount of disorder (closeness to a uniform distribution) increases 430 with the start time of recurrent synaptic learning. The intuition is as follows. The  We observe that, in these last two realizations of the network model (all-to-all 495 connectivity and radius connectivity), the resulting OP distribution is close to uniform; 496 see Fig 7A and Fig 10. Though each orientation has about equal representation in all 497 example networks, the spatial distribution of the cells with each OP changes drastically 498 across each network (recall the maps plotted in Fig 9).

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To see if the choice of radius in calculating the difference in OP greatly affects the 500 results shown in Fig 9,  11. Notice that the measure is low (there is order in the OP map) for small radii, and 502 increases with increasing radius, implying that cells share an OP at small distances, but 503 not at large distances. Importantly, for the networks containing GJ coupling during the 504 first phase of development, and for cases in which the feedforward synapses were 505 allowed to learn for a sufficient amount of time while the GJs are present, there seems 506 to be no order for any value of the radius, see blue and green solid lines in Fig 11. For 507 the same amount of feedforward learning, there is significantly more order for small 508 radii in networks that did not contain GJ coupling, see green dotted curve in Fig 11.

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This further supports our hypothesis the GJ-coupling during the first phase of 510 development serves to enhance disorder in the OP map.  Specifically, we show that the model captures experimentally-measured phenomena 520 such as the preference for cells that were GJ coupled in the first phase of development 521 to share an OP and develop preferentially a bidirectional synapse later in development. 522 In addition to capturing experimentally-observed phenomena, the model also predicts 523 that GJ-coupled cells have a higher firing rate, leading to a faster rate of learning for 524 their LGN synapses when compared to cells that do not have a GJ. We predict that this 525 increased learning rate for sparsely-coupled GJ sister cells, together with the fact that 526 several sets of sister cells are intermingled in the cortex, leads to the formation of a 527 disordered OP map. 528 We investigate this hypothesis by restricting the recurrent synaptic connections of 529 each cortical cell to a small radius and varying the developmental time at which those 530 recurrent synapses begin learning (which is also the time that GJ-coupling between 531 sister cells disappear). We find that the earlier in developmental time that recurrent 532 synapses are allowed to learn, the more ordered the adult OP map. To understand the 533 influence of GJ-coupling between sister cells, we perform the same analysis for networks 534 in which sister cells were not coupled by GJs in the first phase of development. We find 535 that the timing of recurrent synapses still plays a large role in determining order, but 536 that every network without GJ-coupling in the first phase of development leads to an 537 OP map that exhibits more order than the corresponding network that did have GJ 538 coupling. This leads us to conclude that GJ coupling during the first phase of develop 539 indeed promotes a disordered OP map, but works together with the relative timing of 540 synaptic development from LGN and within the cortex.

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The mechanism behind the shared OP of GJ-coupled cells lies in the synchrony (or 542 strongly correlated spike times) induced between the two cells by the GJ. As the LGN 543 synapses form, if the cells are firing synchronously, then those cells will preferentially 544 develop the same set of strengthened LGN synapses, thus forming a similar OP. In our 545 model, sparsity of GJ coupling between the sister cells is essential for this synchrony to 546 occur, and consequently for the shared OP of GJ-coupled cells. When cells are coupled 547 with a probability of 5%, each cell is coupled to an average of 1.5 other cells, leading to 548 isolated pairs or triplets of GJ-coupled cells. As the coupling percentage increases, there 549 are no longer isolated pairs or triplets of cells; rather, each cell may be coupled to postnatal week is about 28%, this was specifically measured for isolated pairs of 557 radially-aligned sister cells [7]. In this work, we are interested in sister cells that are 558 GJ-coupled laterally (within the layer), which hasn't explicitly been measured. 559 Finally, we note that the model developed in this work is highly idealized, especially 560 in its size, spatial structure, and LGN input organization. Regardless, the model