Propagation of orientation selectivity in a spiking network model of layered primary visual cortex
[['Understanding the complex computations performed by neural networks in the nervous system is a central challenge in neuroscience .', 'In recent years , research in rodent sensory systems has provided access to many new facts , but has also raised new theoretical questions .', 'In particular , the role of cortical layers [1 , 2] and other distinct subpopulations [3 , 4] has received increasing attention .', 'While the anatomical path of sensory input from the thalamus through cortical layers has been particularly well characterized for the visual system , it provides little explanation yet how the information is processed and transformed in transit .', 'Here , we apply computational neuroscience techniques to gain new insight into the various computational steps within a highly recurrent neural network .', 'From a theoretical perspective , the cortical network constitutes a dynamical system , which operates on different types of sensory input .', 'To study basic properties of such networks , its input has been assumed to be identical for all neurons [5 , 6] .', 'In a sensory system , however , the input varies across neurons , as sensors extract different aspects of the stimulus .', 'The primary visual cortex ( V1 ) represents an interesting example to study such systems [7 , 8] .', 'In this case , non-homogeneous input for example carries information about the orientation of moving gratings or light bars presented to the eye of an animal [9] .', 'A major difference of rodent visual cortex compared to carnivores and primates is the absence of orientation columns and maps [10] .', 'Although the “salt-and-pepper” organization of orientation preferences implies that lateral interactions are quite unspecific [11 , 12] , neurons nevertheless exhibit responses that are strongly tuned with respect to oriented light bars or gratings [1 , 3 , 13] .', 'Interestingly , this output tuning is already present at eye opening in juvenile mice , where recurrent connectivity appears to be functionally random [14] .', 'In recent theoretical studies , this phenomenon could be explained by a strong attenuation of the untuned component of the distributed input , due to the dominance of inhibition in the recurrent network [7 , 8 , 15–17] .', 'While these previous works identified the mechanism underlying strong output tuning in the primary visual cortex of rodents , they provide no explanation for the different degrees of orientation selectivity in different sub-populations , like cortical layers .', 'In the present work , we apply computational network modeling techniques to study the emergence of orientation selectivity in a layered cortical network model , with a focus on the differences between layers and neuronal subpopulations .', 'As a starting point , we adopt the anatomically and physiologically founded network model developed by Potjans and Diesmann [2] and extend it by orientation selective input similar to Sadeh et al . [8] .', 'We model an experiment in which an animal is shown a sinusoidal grating as a visual stimulus .', 'By this approach , we can focus on the influence of synaptic connectivity on neuronal tuning as all other parameters are chosen to be the same over all populations .', 'Without any adaptation or fine-tuning of parameters , the resulting model matches the distributions of orientation selectivity found in electrophysiological recordings , including the weak tuning of inhibitory neurons as well as L5 pyramidal neurons .', 'We then show that these specific properties cannot be attributed to any specific projection in the circuit .', 'Rather , it is an emergent network feature of the entire recurrent microcircuit .', 'Furthermore , we suggest a concept for the design of novel optogenetic stimulation experiments .', 'Applying this concept to the V1 cortical microcircuit , we can make specific predictions for the outcome of experiments to manipulate the orientation selectivity of L5 principal cells .', 'The analysis of such experiments can eventually confirm or refute our analysis of the circuit .'], ['The network model considered throughout our study is , in essence , the same as the recurrent network model developed by Potjans and Diesmann [2] , extended by thalamic input with a weak orientation bias as described in Sadeh et al . [8] .', 'Potjans and Diesmann [2] created a full-scale neural network model , matching as close as possible the nervous tissue underneath 1 mm2 of neocortex .', 'They incorporated a large set of experimental studies , most prominently the ones by Thomson et al . [18] and Binzegger et al . [19] .', 'They derived a generic connectivity map between eight populations of neurons situated in four layers ( L2/3 , L4 , L5 , L6 ) , one excitatory ( e ) and one inhibitory ( i ) population in each layer ( see Fig 1A for an illustration of the model ) .', 'The model network consists of close to 80 000 leaky integrate-and-fire neurons , where the respective size of individual subpopulations varies strongly across layers .', 'While the two largest populations ( L4e and L2/3e ) contain more than 20 000 neurons , L5e comprizes only about 5 000 neurons ( Fig 1D ) .', 'Throughout all layers , inhibitory populations are considerably smaller than the respective excitatory populations , by a factor of four on average .', 'In our work , the leaky integrate-and-fire neurons are all equipped with delta synapses , i . e . with each incoming spike , the membrane potential is instantaneously deflected by a fixed voltage J , decaying with the membrane time constant τ .', 'The synaptic efficacy J varies depending on the source and target population , respectively .', 'Our model has a fixed number of synapses for each projection , i . e . each neuron of the postsynaptic population P receives the same number of synaptic inputs KPT from the presynaptic population T . For any postsynaptic neuron in the target population , the presynaptic neurons are drawn randomly from the source population .', 'The numbers KPT , which are the essential parameters of our model , were derived by Potjans and Diesmann [2] and are graphically represented in Fig 1B .', 'In addition to recurrent synaptic connections , neurons of the model network receive external input from two different sources ( Fig 1A ) .', 'The first type of input , termed “background input” throughout this paper , is targeting all populations .', 'It represents axons from other cortical and subcortical regions , including gray matter and white matter projections ( Fig 1B/1E ) , but excludes those originating from the thalamus .', 'These background inputs are independent of the stimulus .', 'In all simulations , they are modeled as Poisson processes with rate νbg .', 'The second type of input is represented by thalamocortical projections that originate from the lateral geniculate nucleus ( LGN ) ( Fig 1B/1F ) .', 'In primary visual cortex , these projections are the main source of information about a stimulus .', 'This is the component where we made an important addition to the original model of Potjans and Diesmann [2] .', 'We replaced the unspecific thalamic input with an input that depends in a specific way on the stimulus .', 'Throughout this paper , the stimuli considered were oriented moving gratings , which are commonly used in visual neuroscience .', 'Similar to Sadeh et al . [8] , each neuron in one of the two populations that receive input ( i . e . L4 and L6 ) is randomly assigned a preferred orientation θ^i to represent the tuning of the effective compound input from all pre-synaptic thalamic neurons .', 'Specifically , the input to each neuron depends on the orientation θ of the stimulus and varies according to', 'νith ( θ ) =KPthν0th[1+mcos ( 2 ( θ−θ^i ) ) ] .', '( 1 )', 'Here , KPth is the in-degree of this projection , and ν0th is the mean rate of individual thalamic neurons .', 'We only require the compound input to be tuned and do not make any assumptions about the origin of this tuning .', 'An illustration of the input variation is provided in Fig 1C .', 'Note that orientations vary between 0° and 180° as we do not consider direction selectivity in this study .', 'Similar to the background input , the thalamic input was conceived as a homogeneous Poisson process .', 'It is instructive to analyze and compare two different modes of operation .', 'The stimulation condition is the situation described above , emphasizing the presentation of a structured visual stimulus ( here , a moving oriented grating ) to the animal .', 'In contrast , the spontaneous condition emulates the presentation of a homogeneous gray screen to the animal .', 'In this case , the intensity of thalamic inputs is the same as the rate of the background inputs .', 'Moreover , the angular modulation m is set to zero , removing any information about stimulus orientation from the input .', 'The results of this study are based on three different approximations of the network model .', 'Each of them makes additional assumptions about the network dynamics to achieve different levels of abstraction and simplification .', 'At the same time , all renderings are based on the same connectivity parameters and the same single neuron model .', 'In the following , the three models will be briefly described .', 'In the following , the mathematical model implementations are described in detail .', 'While this documentation allows to fully reproduce the model , it is not essential for the comprehension of the results of the study .', 'In the main text of the manuscript we do not refer to these details , and the reader may choose to directly jump to Results .'], ['The results of the spiking neural network simulations performed in NEST and of the detailed data analysis performed on the simulated data are summarized in Fig 2 , both for the spontaneous ( weak , untuned thalamic input ) and stimulated ( tuned thalamic input ) condition .', 'Note that we do not subtract the spontaneous rates in the stimulated condition for our analysis .', 'As expected , the single neuron firing rates ( Fig 2B/2C ) match the values reported by Potjans and Diesmann [2] .', 'Fig 2D/2E compares the firing rates of the model with experimental values obtained by Niell and Stryker [1] from adult mouse visual cortex .', 'In the spontaneous condition ( Fig 2D ) , the model firing rates are somewhat lower compared to experimental values , but are otherwise in good qualitative agreement with them .', 'We find very low activity in L2/3 and higher rates in the granular and infragranular layers , with the exception of population L6e , where rates are also low .', 'For the stimulus condition , Fig 2E compares the evoked rates of the model with the results of Niell and Stryker [1] .', 'The evoked rates are defined as the maximal rate of the individual neurons over all angles .', 'Therefore , they differ from the median values shown in Fig 2C .', 'In this condition , the model rates vary more strongly across layers as compared to the experimental values .', 'The mean and standard deviation ( SD ) of the pairwise spike-count correlations over all populations are ( 0 . 2 ± 10 . 3 ) ⋅ 10−3 and ( 0 . 3 ± 10 . 4 ) ⋅ 10−3 for spontaneous and stimulated conditions , respectively ( 10 ms bin size ) .', 'The mean and SD of the coefficient of variation ( CV ) of the inter-spike intervals are 0 . 9 ± 0 . 1 for both conditions .', 'Both measures show only small variations over different populations ( see S1 and S2 Figs for population-wise quantification ) .', 'The low correlation and Poisson-like irregularity indicate that the network indeed operates in the asynchronous irregular regime [6 , 28] , as illustrated by the raster plot in Fig 2A .', 'The activity of the network is robust to changes in important parameters which are not well constrained by experiments .', 'Although average firing rates vary across parameter sets , their distribution remained qualitatively similar ( S4A/S4B–S9A/S9B Figs ) .', 'Only for very low background input , the network operates in a more synchronous regime with increased correlations ( S3B/S3C and S6D Figs ) .', 'In this case , the network activity also changes qualitatively .', 'Surprisingly , this is only the case for reduced background input .', 'For a two-fold increase , the activity remains stable ( S3B and S5D Figs ) .', 'Our model extends the work of Potjans and Diesmann by giving the thalamic input a weak orientation bias , as described previously [7 , 8] .', 'The macroscopic connectivity pattern of the network ( Fig 1A ) already gives a rough idea of the signal flow through the network .', 'Thalamic afferents project mainly to L4 and to a smaller extent also to L6 .', 'From L4 , the signal is transmitted to L2/3 , which is thought to be the central processing unit of the microcircuit .', 'Signals are then forwarded to higher brain areas as well as to L5 , which is the major output unit for projections to sub-cortical brain areas .', 'In order to dissect the computations performed by the network , we extract tuning curves of all neurons in the model network .', 'As expected from previous studies [7 , 8 , 16 , 17] , during stimulation , the weak orientation bias of the thalamic input is amplified to strongly orientation selective responses in the input population ( Fig 3 ) .', 'Interestingly , L2/3e also shows strong tuning although these neurons only receive input via random connections from L4 neurons .', 'This phenomenon was previously described by Hansel and van Vreeswijk [7] , where projections from L4e to L2/3 were studied independently of other parts of the network .', 'The strong tuning in L2/3e can be explained by a recurrent tuning amplification in inhibition dominated networks [7 , 8] .', 'This effect is enough to create strong output tuning from a small bias in the input , which in turn is due to random sampling of preferred orientations in L4 .', 'In this context it may seem even more surprising that orientation selectivity of L5 neurons appears to be weaker compared to the other layers .', 'This finding , which is a direct consequence of the underlying experimentally determined connectivity , is consistent with several measurements of orientation selectivity performed in different rodent species [1 , 3 , 29–31] ( Fig 4D ) .', 'However , it cannot be easily explained by visual inspection of the connectivity matrix .', 'Even though synaptic connectivity is stronger for the projection to L2/3e , L5e neurons also receive a large fraction of their input from L4e ( Fig 1B , Table 3 ) .', 'Therefore , as the output tuning of L2/3e neurons is essentially inherited from these neurons , one might expect a stronger tuning also in L5e .', 'In order to understand in more detail the differences in orientation selectivity between layers , we extracted the orientation selectivity index ( OSI ) of all neurons in the model network ( Fig 4B/4C ) .', 'We employ a measure based on the circular variance to quantify orientation selectivity ( see Methods for details ) .', 'For each neuron , the OSI is calculated as the ratio of first ( F1 ) and zeroth ( F0 ) Fourier component of the tuning curve ( Fig 4A ) .', 'A stronger modulation of the firing rate over angles ( higher F1 ) , leads to a higher OSI , while a higher mean rate ( higher F0 ) reduces the OSI .', 'Experimental studies often use a different measure for orientation selectivity based on the activity at the preferred and orthogonal orientation .', 'To allow direct comparison with these works , we also calculated this alternative quantity ( S10 Fig ) .', 'While the absolute numbers are generally higher for this measure , the qualitative results are identical .', 'Note that also the distribution of orientation tuning is quite robust against variations in parameters ( S4C–S9C Figs ) .', 'Consistent with experimental findings and throughout all layers , inhibitory neurons have a lower OSI than the corresponding excitatory neurons [1 , 3 , 32] ( Fig 4D ) .', 'While this might be expected for L4 and L6 due to the lower number of thalamic afferents , it is not at all obvious for L2/3 .', 'In this population , it is a consequence of the recurrent network connectivity , as our analysis will show .', 'Note that during spontaneous activity , neurons also exhibit weak apparent orientation selectivity ( Fig 4B ) .', 'This is essentially due to fluctuations in the spiking process .', 'Neurons may randomly fire a few action potentials more for one orientation as compared to another , which results in weak but nonzero orientation selectivity .', 'The magnitudes found here are consistent with a Poisson process with the same rate and recording time .', 'The orientation selectivity is measured as the ratio of first and zeroth Fourier component of the tuning curves ( Fig 4A ) .', 'Analyzing the tuning curves in Fig 3A in more detail , we see that although the F1 component in L5e is stronger than in L2/3e , this is overcompensated by the high F0 component of this population , leading to weaker tuning in L5e .', 'This raises two questions: First , which projections onto L2/3e and L5e lead to the observed difference in activity ?', 'Second , why is the tuned ( F1 ) component of L5e not as strongly amplified as the mean rate ( F0 ) , similar to the situation in L2/3 ?', 'In the following , we will address these questions , employing suitable mathematical and computational methods .', 'While the first question can be answered by analyzing the input currents received by the different populations , the second question requires an approach which takes the strongly recurrent nature of the circuit across layers into consideration .', 'Each neuron in the eight populations of the network receives inputs from several pre-synaptic populations , possibly all with different tuning curves , and forms its own output tuning from those ( Fig 4A ) .', 'It has been shown in experiments that this scenario also reflects the situation in rodents [11] .', 'Which projections are most potent for driving the the target neuron is not immediately obvious from the anatomical connectivity between neurons ( Fig 1B ) .', 'The activity of pre-synaptic neurons is of course also relevant for the total input current to a given neuron .', 'In Fig 5A , the mean input current ( Eq 3 ) for each projection between populations is shown .', 'While the difference between inputs to L2/3e and L5e seems insignificant when looking at the underlying connectivity ( Fig 1B ) , the picture changes completely if the activity of pre-synaptic neurons is taken into consideration ( Fig 5A ) .', 'The input to L2/3e neurons is mostly determined by L2/3i , both L4 populations and the constant background .', 'Input to L5e , on the other hand , is dominated by L5i and background input , but only to a lesser extent by inputs from L2/3i , L4e and L5e .', 'As a result , due to the different excitation-inhibition ( EI ) balance , the net currents imply a higher mean input to L5e compared to L2/3e ( Fig 5C ) .', 'The marked difference between anatomically defined connectivity ( Fig 1B ) and the total input current in the stationary balanced state ( Fig 5A ) highlights the contribution of activity dynamics in recurrent networks [6 , 33 , 34] .', 'While the situation for L5e and L2/3e is very similar when counting the number of excitatory and inhibitory synapses that terminate in each population , the actual current drive they receive is quite different .', 'Since the activity of the whole network autonomously settles in an operating point where the EI balance is dynamically maintained , this can have very different consequences for the various populations , irrespective of the anatomical connectivity .', 'Having demonstrated how the dynamic equilibrium and the resulting operating point of the layered network explains the high firing rate of L5 neurons , we can now ask the same question with regard to orientation tuning in each neuronal population .', 'In combination , these two aspects fully determine the orientation selectivity index ( OSI ) of all neurons .', 'To achieve this , we calculate for each neuron a tuning vector summarizing the information about the stimulus conveyed by the input currents to that neuron .', 'Its magnitude measures the tuning strength of the combined input to the post-synaptic neuron , whereas its direction indicates its preferred orientation ( Fig 5F inset , Eq 4 ) .', 'The total tuning input from one specific source population to a single neuron can then be calculated as the sum of all tuning vectors of neurons in the pre-synaptic population which connect to the post-synaptic neuron .', 'If all tuning curves are cosine-like , this concept exactly corresponds to linear summation of inputs .', 'The magnitudes of the tuning vectors are identical to the first Fourier component ( F1 ) of the tuning curves for the current input .', 'Fig 5B summarizes the mean lengths of all projection tuning vectors , summarizing the information flow in the system .', 'The tuning information enters the cortical network in L4 and L6 and then spreads to the other populations .', 'Although both L4 and L6 neurons show significant tuning to stimulus orientation , due to the low firing rates of L6e neurons it is mostly L4 which provides tuning information to the other populations .', 'In contrast to what would be expected from the connectivity alone ( Fig 1B ) , also L5e neurons receive most of the tuned current from L4 .', 'Comparing the input to L2/3e and L5e neurons in more detail reveals that L2/3e neurons mostly integrate tuned input from both populations in L4 , while L5e neurons receive less tuned input almost exclusively from L4e .', 'Considering only the output modulation of the different populations ( Fig 5D ) , which is defined as the first Fourier component of the tuning curves ( Fig 4A ) , it surprises that despite the less tuned input to L5e , these neurons still show a stronger output modulation than L2/3e .', 'This highlights that , besides the input strength , the input sensitivity of neurons is relevant as well .', 'However , the large modulation component of the output of L5e neurons cannot compensate their high firing rates , leading to a lower OSI ( Fig 5C–5E ) .', 'Comparing the distribution of firing rates and output modulations over the eight populations ( Fig 5C/5D ) , it is evident that these two components are subject to different gain factors .', 'In the following , these dependencies will be studied in more detail .', 'Our analyses so far disentangled the input to neurons in different populations , which explains the observed features of neuronal output .', 'However , the multi-population model is highly recurrent .', 'Therefore , it is not yet clear why the input to all neurons settles in the observed operating point .', 'We face a chicken-and-egg problem here , as in the recurrent network , the output of all populations is simultaneously the input to the same populations , eventually settling in the dynamical operating point .', 'In order to tackle this problem , we now change our perspective from analyzing the input-output relation of single neurons to input-output behavior of the entire network .', 'The idea is to manage the problem by considering the network as a distributed system , which can perform its function only as a whole .', 'This approach is best realized by linearizing the network dynamics about its operating point .', 'We obtain our close to 80000-dimensional linearized network model ( Model C ) by first formulating a single neuron firing rate model based on the single neuron transfer function Fi ( ν , νth ) ( Model B ) .', 'The firing rate model is then linearized about its dynamical operating point , leading to the explicit input-output relation', 'Δν= ( 1−W ) −1Δβ .', '( 13 )', 'Here , Δν is a vector summarizing all single neuron rate changes due to a change in the effective input Δβ , which in turn is the change in thalamic input rate scaled by the input sensitivities of the different populations .', 'These sensitivities can also be understood as the feed-forward gains of the system [8] , see Methods for details .', 'Furthermore , the behavior of the recurrent model is governed by the matrix W , which is the matrix of effective recurrent connections , which are a product of the anatomically and physiologically defined synaptic weights and the input-output sensitivities ( gains ) of individual post-synaptic neurons at their operating point .', 'In combination with the activity at the operating point ν0 , the rate change Δν results in the output activity ν = ν0 + Δν of the linear network model .', 'The explicit form of Eq 13 also supports the idea of considering the network as an integrated system with one single vector-valued input-output relation .', 'Importantly , this relation is governed by the inverse of the effective connectivity 1−W .', 'Because the entries in Wij are a function of Kij , this inversion potentially distributes the influence of each single connectivity parameter Kij over the entire network .', 'This observation also explains why the effects of individual connectivity parameters can be counterintuitive [35] .', 'The firing rate model ( Model B ) as well as the linear network model ( Model C ) can be considered as simplifications of the spiking neural network ( Model A ) , as they make additional assumptions about its dynamics .', 'Therefore , before analyzing these models in more detail , it is important to establish the consistency of their respective behavior .', 'S11 Fig compares the results from a simulation of all three models , for matched network parameters .', 'We found that the single neuron firing rates of the three models are in good agreement .', 'While the similarity between the non-linear rate model ( Model B ) and the linear model ( Model C ) is quite high , both exhibit mild discrepancies from the spiking network model ( Model A ) .', 'In particular , for the larger firing rates of L5e , the non-linear rate model and the linear model tend to slightly underestimate them .', 'Generally , also the preferred orientations and orientation selectivities are in good agreement between the different models .', 'While the match is again excellent for the non-linear firing rate model and linear network model , both slighly deviate from the spiking network model .', 'This can be explained by the role played by activity fluctuations in this model .', 'In particular , the observed orientation selectivities are somewhat higher here , reflecting the same positive bias as the one observed during spontaneous activity .', 'The match between the three models is also very robust with respect to changes in parameters ( S4A/S4B/S4C–S9A/S9B/S9C Figs ) .', 'In fact , even for reduced background input , when there is substantial synchrony in the spiking activity , the non-linear rate and linear models still provide reasonable approximations ( S6A/S6B/S6C Fig ) .', 'Now that the general consistency in the behavior of the different models is established , we can analyze the linearized network and be confident that our conclusions are also valid for the spiking network .', 'In previous work , a separation into two separate , non-interfering pathways were postulated , exploiting that the network can have different gains for different components of its input [8] ( Fig 6A ) .', 'Specifically , the effective input perturbation is split into a baseline and a modulation component such that Δβ = ΔβB + ΔβM .', 'Here , we generalize this decomposition to our present multi-population model .', 'Furthermore , we present a novel analytical approach for their analysis .', 'For each population , the baseline ΔβB is conceived as the mean input these neurons receive during stimulation .', 'Since the mean input is different for each population , this leads to a population-wise constant input vector ( Fig 6B , red curve ) .', 'The modulation ΔβM , in contrast , accounts for all neuron-by-neuron deviations from the population baseline , ΔβM ≔ Δβ − ΔβB , yielding a vector with zero mean in each of the eight populations ( Fig 6B , inset ) .', 'Most importantly , the modulation component includes the tuning information that each neurons receives , based on the neuron-specific orientation bias of the stimulus .', 'In other words , the deviation of ΔβM from the baseline is determined by the orientation of the stimulus , as described by Eq 6 .', 'Note that the baseline and modulation components are both only indirectly related to mean rate ( F0 ) and tuning strength ( F1 ) discussed in the previous section .', 'While the latter are calculated for the tuning curve of each single neuron independently , the baseline and modulation of the input are calculated over all neurons in one population , for a single stimulation angle .', 'Separation of baseline and modulation pathway means that the two input components ΔβB and ΔβM are processed by two independent mechanisms with possibly distinct gains γB and γM .', 'As shown in Methods , the two pathways are governed by', 'ΔνB= ( 1−Q ) −1ΔβBΔνM= ( 1−S ) −1ΔβM ,', 'where the decomposition W = Q + S was used .', 'Thereby , Q contains population-wise expectation values , resulting in an 8 × 8 block structure , and S are the individual modulations S = W − Q of single connections in the network .', 'Importantly , there is no cross-talk between the two pathways .', 'A change in the input baseline ΔβB does not lead to change in the output modulation ΔνM , and vice versa .', 'Solving these two systems independently yields two output rate changes ΔνB and ΔνM .', 'If the cross-terms in the calculations are indeed negligible , the solution Δν = ΔνB + ΔνM provides a good approximation to the network behavior ( see Eq 12ff ) .', 'For the network under study here , the mean magnitudes μ of the cross-terms are μ[|QΔνM|] = 0 . 02 and μ[|SΔνB|] = 0 . 09 as compared to μ[|Δβ|] = 3 . 63 , implying that the system can indeed be studied by treating the baseline and the modulation pathway separately .', 'Similar numbers are also obtained for the other parameter sets ( S3E Fig ) .', 'The solutions of the baseline and modulation systems , ΔνB and ΔνM , are shown in Fig 6C in comparison to the solution of the W-system .', 'Throughout all populations , the separation solution is in excellent agreement with the direct solution .', 'The modulation part , which conveys the tuning information of the neurons , matches almost perfectly ( Fig 6C inset ) .', 'The good agreement is further confirmed by the high coefficient of determination ( “variance explained” ) of R2 = 99 . 1% .', 'For the different parameter sets , the lowest R2 is obtained for increased background input with R2 = 98 . 0% ( S3D , S4E–S9E Figs ) .', 'This demonstrates the robustness of the baseline and modulation decomposition also with respect to changes in parameters .', 'In previous sections , we identified the high baseline rate of L5e as the main cause for the low orientation selectivity of its neurons .', 'This becomes manifest in a high baseline output rate ΔνB for this particular population ( Fig 6C ) .', 'While this observation still does not fully reveal the underlying reason for the high rates , it hints at the effective mean connectivity represented by Q . In the following , we will therefore study the baseline pathway in more detail .', 'We start by summarizing the results of Sadeh et al . [8 , 36] , where a similar scenario for a two population EI network was studied .', 'In analogy to the present model , the two population system can also be described by a linear system of the form', 'Δν= ( 1−W ) −1Δβ ,', 'where the dimensionality of the system equals the total number of neurons .', 'In order to study the network behavior , it is instrumental to inspect the eigenvalue spectrum of the matrix W . In the two population case , when both populations receive the same input , the eigenvalue spectrum consists of a bulk of known radius localized at the origin and a single exceptional eigenvalue λ [37] .', 'For inhibition dominated networks , this exceptional eigenvalue is real and negative ( Fig 7A ) .', 'Furthermore , the eigenvector Ψ corresponding to that exceptional eigenvalue is the uniform vector , with all entries identical .', 'In this two-population network model , the baseline component of the input perturbation is also proportional to the uniform vector', 'ΔβB∝Ψ∝ ( 1⋮1 ) .', 'Therefore , exploiting the eigenvector property of a baseline perturbation , the exceptional eigenvalue can be transformed into the gain factor of the baseline', 'ΔνB= ( 1−W ) −1ΔβB= ( 1−λ ) −1ΔβB=λ˜ΔβB .', 'For inhibition dominated networks , this gain factor λ˜ has a s
['Introduction', 'Methods', 'Results', 'Discussion']
['Neurons in different layers of sensory cortex generally have different functional properties .', 'But what determines firing rates and tuning properties of neurons in different layers ?', 'Orientation selectivity in primary visual cortex ( V1 ) is an interesting case to study these questions .', 'Thalamic projections essentially determine the preferred orientation of neurons that receive direct input .', 'But how is this tuning propagated though layers , and how can selective responses emerge in layers that do not have direct access to the thalamus ?', 'Here we combine numerical simulations with mathematical analyses to address this problem .', 'We find that a large-scale network , which just accounts for experimentally measured layer and cell-type specific connection probabilities , yields firing rates and orientation selectivities matching electrophysiological recordings in rodent V1 surprisingly well .', 'Further analysis , however , is complicated by the fact that neuronal responses emerge in a dynamic fashion and cannot be directly inferred from static neuroanatomy , as some connections tend to have unintuitive effects due to recurrent interactions and strong feedback loops .', 'These emergent phenomena can be understood by linearizing and coarse-graining .', 'In fact , we were able to derive a low-dimensional linear dynamical system effectively describing stimulus-driven activity layer by layer .', 'This low-dimensional system explains layer-specific firing rates and orientation tuning by accounting for the different gain factors of the aggregate system .', 'Our theory can also be used to design novel optogenetic stimulation experiments , thus facilitating further exploration of the interplay between connectivity and function .']
['Understanding the precise roles of neuronal sub-populations in shaping the activity of networks is a fundamental objective of neuroscience research .', 'In complex neuronal network structures like the neocortex , the relation between the connectome and the algorithm implemented in it is often not self-explaining .', 'To this end , our work makes three important contributions .', 'First , we show that the connectivity extracted by anatomical and physiological experiments in visual cortex suffices to explain important properties of the various sub-populations , including their selectivity to visual stimulation .', 'Second , we introduce a novel system-level approach for the analysis of input-output relations of recurrent networks , which leads to the observed activity patterns .', 'Third , we present a method for the design of future optogenetic experiments that can be used to devise specific stimuli resulting in a predictable change of neuronal activity .', 'In summary , we introduce a novel framework to determine the relevant features of neuronal microcircuit function that can be applied to a wide range of neuronal systems .']
['action potentials', 'medicine and health sciences', 'neural networks', 'membrane potential', 'brain', 'electrophysiology', 'neuroscience', 'mathematics', 'algebra', 'network analysis', 'computational neuroscience', 'neuronal tuning', 'computer and information sciences', 'animal cells', 'visual cortex', 'cellular neuroscience', 'cell biology', 'linear algebra', 'anatomy', 'single neuron function', 'neurons', 'physiology', 'biology and life sciences', 'cellular types', 'physical sciences', 'computational biology', 'eigenvalues', 'neurophysiology']