Visually evoked signals in the retina pass through the dorsal geniculate nucleus (dLGN) on the way to the visual cortex. This is however not a simple feedforward flow of information: there is a significant feedback from cortical cells back to both relay cells and interneurons in the dLGN. Despite four decades of experimental and theoretical studies, the functional role of this feedback is still debated. Here we use a firing-rate model, the extended difference-of-Gaussians (eDOG) model, to explore cortical feedback effects on visual responses of dLGN relay cells. For this model the responses are found by direct evaluation of two- or three-dimensional integrals allowing for fast and comprehensive studies of putative effects of different candidate organizations of the cortical feedback. Our analysis identifies a special mixed configuration of excitatory and inhibitory cortical feedback which seems to best account for available experimental data. This configuration consists of (i) a slow (long-delay) and spatially widespread inhibitory feedback, combined with (ii) a fast (short-delayed) and spatially narrow excitatory feedback, where (iii) the excitatory/inhibitory ON-ON connections are accompanied respectively by inhibitory/excitatory OFF-ON connections, i. e. following a phase-reversed arrangement. The recent development of optogenetic and pharmacogenetic methods has provided new tools for more precise manipulation and investigation of the thalamocortical circuit, in particular for mice. Such data will expectedly allow the eDOG model to be better constrained by data from specific animal model systems than has been possible until now for cat. We have therefore made the Python tool pyLGN which allows for easy adaptation of the eDOG model to new situations.
Visually evoked signals pass the dorsal geniculate nucleus (dLGN) on the route from retina to primary visual cortex in the early visual pathway. This is however not a simple feedforward flow of information, as there is a significant feedback from primary visual cortex back to dLGN. Cortical cells feed back to both relay cells and interneurons in the dLGN, and also to cells in the thalamic reticular nucleus (TRN) which in turn provide feedback to dLGN cells [1,2]. In the last four decades numerous experimental studies have provided insight into the potential roles of this feedback in modulating the transfer of visual information in the dLGN circuit [3–19]. Cortical feedback has been observed to switch relay cells between tonic and burst response modes [20,21], increase the center-surround antagonism of relay cells [16,17,22,23], and synchronize the firing patterns of groups of such cells [10,13]. However, the functional role of cortical feedback is still debated [2,24–30]. Several studies have used computational modeling to investigate cortical feedback effects on spatial and/or temporal visual response properties of dLGN cells [31–38,53]. These have typically involved numericallyexpensive dLGN network simulations based on spiking neurons [31–33,35,38] or models where each neuron is represented as individual firing-rate unit [36,37]. This is not only computationally cumbersome, but the typically large number of model parameters in these comprehensive network models also makes a systematic exploration of the model behavior very difficult. In the present study we instead use a firing-rate based model, the extended difference-of-Gaussians (eDOG) model [39], to explore putative cortical feedback effects on visual responses of dLGN relay cells. A main advantage with this model is that visual responses are found from direct evaluation of two-dimensional or three-dimensional integrals in the case of static or dynamic (i. e. , movie) stimuli, respectively. This computational simplicity allows for fast and comprehensive study of putative effects of different candidate organizations of the cortical feedback. Taking advantage of the computational efficiency of the eDOG model, we here explore effects of direct excitatory and indirect inhibitory feedback effects (via dLGN interneurons and TRN neurons) on spatiotemporal responses of dLGN relay cells. In particular we investigate effects of (i) different spatial spreads of corticothalamic feedback and (ii) different corticothalamic propagation delays. Our analysis suggests that a particular mix of excitatory and inhibitory cortical feedback agrees best with available experimental observations. In this configuration an ON-center relay cell receives feedback from ON-center cortical cells (ON-ON feedback), consisting of a slow (long-delay) and spatially widespread inhibitory feedback combined with a fast (short-delay) and spatially narrow excitatory feedback. Here the inhibitory and excitatory ON-ON feedback connections are accompanied by excitatory and inhibitory OFF-ON connections, respectively, following a phase-reversed arrangement [38]. For one this feedback organization accounts for the feedback-induced enhancement of center-surround antagonism of relay cells as observed in experiments [16,17,22,23,38]. Further, it seems well suited to dynamically modulate both the center-surround suppression and spatial resolution, for example, to adapt to changing light conditions [40]. Moreover, a longer thalamocortical loop time of ON-ON inhibitory feedback loop compared to ON-ON excitatory feedback may contribute to temporal decorrelation of natural stimuli [41], an operation that has been observed accomplished at the level of dLGN in the early visual pathway [42]. At the same time, the rapid excitatory feedback may contribute to linking stimulus features by synchronizing firing of neighboring relay cells [10,19]. Previous experimental studies have focused on cat, monkey, and ferret dLGN, and the present model was adapted to neurobiological findings from cat. However, the last years have seen a surge of interest in mouse visual system, where new optogenetic and pharmacogenetic methods provide new tools for precise manipulation of identified neurons in the thalamocortical circuit [43–48]. Such data will expectedly allow for a detailed adaptation of the eDOG model to rodent dLGN, likely much better constrained by biological findings than what has been possible until now for cat. To facilitate this we have made the Python tool pyLGN (http: //pylgn. rtfd. io) which allows for easy modification and evaluation of the eDOG model to new situations.
Spike responses of neurons in the early visual pathway are most commonly described in terms of receptive fields. Mathematically, the spatiotemporal receptive field is defined by an impulse-response function W (r, t). This function describes the firing-rate response to a tiny (δ-function) spot positioned at r = 0 which is on for a very short time (δ-function) at t = 0. If linearity is assumed, the response to any stimulus S (r, t) can be found by convolving the impulse-response function with the stimulus [39,49–52]: R (r, t) =∫τdτ ∫∫r′d2r′ W (r−r′, τ) S (r′, t−τ), (1) or written more compactly R (r, t) = W * S. (2) Here S (r, t) is a spatiotemporal stimulus function describing, e. g. , the light intensity on a screen as a function of time and position. R (r, t) is the response of a neuron with its receptive-field center at r. The spatial integral goes over the whole visual field, i. e. , over all two-dimensional space. For mathematical convenience we have chosen the temporal integral to go from τ = −∞ to + ∞. Since a stimulus input cannot affect the response in the past, it then follows that W (r, τ < 0) = 0. In Fourier space the convolution in Eq (2) corresponds to a product R ˜ (k, ω) = W ˜ (k, ω) S ˜ (k, ω), (3) where R ˜, W ˜, and S ˜ are the Fourier transforms of the neural response R, the impulse-response function W, and the stimulus S, respectively. The tilde symbol (∼) will be used to denote the Fourier transform of any function throughout this paper. The function argument k is the wave vector which is related to the spatial frequency ν via |k| = 2πν. Correspondingly, the angular frequency ω is related to the temporal frequency f via ω = 2πf. With W ˜ and S ˜ known, the neural response can thus always be found by an inverse Fourier transform F - 1 { }, which entails an integral over temporal and spatial frequencies R (r, t) = F - 1 { W ˜ (k, ω) S ˜ (k, ω) }. (4) The response model in Eq (1) is an example of a descriptive model where the purpose is to summarize experimental data compactly in a mathematical form [51–53]. Here the aim is to find an appropriate impulse-response function, i. e. , spatiotemporal receptive-field function, that describes the measured neural response to different visual stimuli [50]. With this approach, however, limited insight is gained into how the neurons and neural circuitry in the early visual system provide such a receptive field. To address this question a mechanistic receptive-field model is needed. For a discussion of the difference between descriptive and mechanistic models in visual neuroscience, see [53,54]. In mechanistic LGN-circuit models the input from retinal ganglion cells have been described by descriptive models, see, e. g. , [36,37,39,52,55]. Likewise, in the present eDOG model the input from retinal ganglion cells is represented by the descriptive impulse-response function (Eq (1) ). Here a square grid of retinal ganglion cells with identical, spatially-localized receptive fields are considered (see Fig 1). The activity, i. e. , firing rates, of the neurons on the retinal ganglion cell layer then serves as input to the dLGN relay cell layer. This is represented by a spatiotemporal coupling-kernel function KRG, which reflects the direct synaptic input from retinal ganglion cells to dLGN relay cells. The coupling kernel, which is analogous to the descriptive impulse-response function in Eq (1), is assumed to only depend on the relative distance between the cells in the visual field [52]. The response of a relay cell located at r is then given by [39,52]: RR (r, t) =∫τdτ ∫∫r′d2r′ KRG (r−r′, τ) RG (r′, t−τ) =KRG*RG, (5) where RR and RG are the firing-rate responses of relay cells and ganglion cells, respectively. The coupling kernel KRG (r − r′, τ) denotes the strength with which the response of a ganglion cell, displaced by r − r′ from the relay cell, at time t − τ influences the response of the latter at time t. Note that, KRG (r, τ < 0) = 0 due to causality. In Fourier space the relationship in Eq (5) can be written as W ˜ R S ˜ = K ˜ R G W ˜ G S ˜, (6) where we have used the general relationship in Eq (3). The key point here is that a descriptive model for the relay-cell impulse-response function W ˜ R now has a mechanistic interpretation. This relation is given as the product of the impulse-response function W ˜ G of the retinal ganglion cells and the coupling kernel K ˜ R G from the former cell type to the latter. In the eDOG model this approach is extended to include the various feedforward and feedback connections affecting the relay-cell response. The result is an expression for the relay-cell impulse-response function W ˜ R in terms of the impulse-response function W ˜ G of the retinal input and the coupling kernels connecting the neurons of the circuit. With such a mechanistic expression for W ˜ R, the response to any visual stimulus can be computed by means of the inverse Fourier transform in Eq (4). Here we derive the impulse-response function for dLGN relay cells for the mechanistic eDOG model [39]. The complete circuit is shown in Fig 2. In this figure each cell type correspond to a two-dimensional layer (or population) of identical cells. We will in the following focus on the dLGN relay cells with ON symmetry, but a similar model can be constructed for OFF-symmetry cells. These neurons receive feedforward excitation and indirect feedforward inhibition (via intrageniculate interneurons) from ON-center ganglion cells in retina. The relay cells further receive cortical feedback from both cortical ON cells and cortical OFF cells. In order to allow for easy exploration of the eDOG model and in particular effects of cortical feedback on relay-cell responses, we have developed an efficient, firing-rate based simulator of spatiotemporal responses in the early visual system. The simulator is named pyLGN and is written in Python. The design goals for pyLGN are to provide a software framework for studying the cortical feedback effects that is easy to use, extensible, and open. To facilitate usability, pyLGN has its own documentation page including installation instructions, several usage examples, and technical aspects (http: //pylgn. rtfd. io). To achieve extensibility, object-oriented programming is used, making it possible for the user to define new connectivity kernels and input stimuli. Lastly, to support openness pyLGN is both open-source and multi-platform. All calculations presented in this paper have been tracked using the Python software Sumatra [76], which is an automated tracking tool for computational simulations and analysis. The source code for all presented simulations is available at (https: //github. com/miladh/edog-simulations).
As a first example application we focused on the effect of cortical feedback on the spatial response properties of dLGN cells. A specific focus was on so-called area-response curves, i. e. , responses to circular spots and patch-gratings as a function of stimulus size, which has received substantial experimental attention. In particular, studies have reported several effects of cortical feedback including sharpening of the receptive field by enhancing the center-surround antagonism of relay cells, increased receptive field center size with removal of feedback, and increased peak response to a an optimal diameter stimulus [17,22,23]. Other studies have reported a more diverse influence from corticothalamic feedback, including both facilitatory and suppressive effects on dLGN cell responses, and changes in the receptive-field structure [18]. Further, several studies have indicated that cortical feedback plays a role in extraclassical suppression in LGN [8,98,99]. A recent study confirms that extraclassical suppression indeed involves extraretinal mechanisms with a delay that is consistent with polysynaptic inhibition [100]. However, the early phase of the suppression is unlikely to involve cortical feedback, as the onset of the suppression is too fast, even though the cortical feedback may contribute in later phases of suppression. Our model demonstrated that cortical feedback can, depending on the feedback configuration, both enhance and suppress center-surround antagonism and both increase and decrease the receptive-field center size of relay cells. While the receptive-field center size decreases and the center-surround antagonism (as measured by the suppression index αs) increases with increased (indirect) cortical inhibitory feedback, the opposite is seen for excitatory feedback (Fig 7). These results support that a phase-reversed arrangement of the cortical feedback, where the ON-ON feedback is inhibitory while the OFF-ON feedback is excitatory, as suggested by data from [73], is more effective to enhance the center-surround antagonism of relay cells as observed in experiments [16,17,22,23,38]. However, with this arrangement a reduction in response to the optimal diameter stimulus in the size tuning curves was observed in our model (Fig 7), in contrast to some experimental studies where an increase in response was reported [17] Here we also considered the more complicated mixed phase-reversed feedback situation with a spatially broad ON-ON inhibitory feedback (combined with a corresponding OFF-ON excitatory feedback) and a spatially narrow ON-ON excitatory feedback (combined with a corresponding OFF-ON inhibitory feedback). Such a center-surround spatial organization of the feedback with excitatory bias to center and an inhibitory bias to the surround has been seen experimentally [3,16]. In our model studies such mixed feedback was seen to give increased center-surround antagonism compared to the situation with ON-ON inhibitory feedback alone. Further, this configuration could also both reduce the size of the optimal stimulus diameter, as well as increase the magnitude of the response to the optimal stimulus diameter (Fig 7). Correspondingly, with this configuration a sharper band-pass property of the spatial-frequency spectra was observed (Fig 8). As for the spatial response properties, the effects of ON-ON inhibitory and ON-ON excitatory feedback (accompanied by the corresponding phase-reversed OFF-ON feedback) are seen to be quite distinct. While delayed inhibitory feedback makes the impulse response more biphasic, the opposite is the case for delayed excitatory feedback (Fig 11). Likewise, while the temporal frequency tuning becomes sharper with delayed inhibitory feedback, it becomes blunter with delayed excitatory feedback (Fig 13, top row). These features, i. e. , increased biphasic index and sharper temporal frequency tuning, are maintained also for the case of mixed cortical feedback as long as the thalamocortical loop delay for the inhibitory feedback is much larger than for the excitatory feedback (Figs 12 and 13, bottom row). Such a mixed-feedback configuration is also found to be particularly suited to remove temporal correlation in the stimulus and thus reduce the temporal redundancy in the neural signals that are sent from dLGN relay cells to cortex (Fig 14). Our results concerning the spatial and temporal feedback effects suggests that a situation with a mixed organization of cortical feedback consisting of a slow (long-delay) and spatially widespread ON-ON inhibitory feedback, combined with a fast (short-delay) and spatially narrow ON-ON excitatory feedback may have particular advantages. Here the inhibitory and excitatory ON-ON feedback connections are accompanied by excitatory and inhibitory OFF-ON connections following a phase-reversed arrangement [38]. This specific prediction of a mixed organization of feedback could be tested experimentally, for example, by pharmacological inactivation (or other means) of specific feedback connections in the circuit. This feedback organization seems well suited to dynamically modulate both the center-surround suppression and spatial resolution, for example, to adapt to changing light conditions where the most efficient neural representation of the stimulus is expected to vary depending on the signal-to-noise ratio [40]. In particular, for high light levels (i. e. , high signal-to-noise) a band-pass like spatial spectrum (as obtained with our model for certain parameter choices) is expected to provide the most efficient coding, while for low light levels (low signal-to-noise) a low-pass spatial spectrum (as obtained with our model for some other parameter choices) seems better (see Sec. 3. 6. 1 in [101]) Further, a longer thalamocortical loop time of ON-ON inhibitory feedback compared to ON-ON excitatory feedback assures that temporal correlations in the natural visual stimuli are reduced in the relay-cell responses (Fig 14). This temporal feedback arrangement gives a large biphasic index (Fig 12) which previously has been shown to provide temporal decorrelation of natural stimuli [41], a feature that has also been seen in experiments [42]. An increased biphasic index in the presence of cortical feedback, has also been seen in other modeling studies [37,102]. In particular, a study using a predictive coding model, where the phase-reversed configuration emerged during training, reported that stronger biphasic response in LGN may result from (predictive) cortical feedback interactions [102]. While the slow inhibitory feedback is key for providing temporal decorrelation, the rapid excitatory feedback may have a role in linking stimulus features by synchronizing firing of neighboring relay cells to provide a strong input to cortical target cells [10,19]. Interestingly, a recent study found a large variation in axonal conduction times for corticothalamic axons, from a few milliseconds to many tens of milliseconds [84]. This suggests that differences in feedback delays indeed may have a functional role. It should be noted that the eDOG-model [39] on which pyLGN is based, assumes that the cortical feedback has a phase-reversed arrangement where each ON-ON feedback connection is accompanied by a phase-reversed OFF-ON feedback connection (Eq (11) ), i. e. , a push-pull arrangement as experimentally observed in [73]. An alternative is a phase-matched arrangement where relay cells receive feedback only from cortical cells with the same symmetry, including both the direct excitatory feedback and the indirect inhibitory feedback. However, such an arrangement is not only at odds with the observations in [73], but also fails to explain the experimentally observed cortical-feedback induced increase in center-surround antagonism [38]. The aim of present work has been to introduce the simulation tool pyLGN and to provide a variety of example results to illustrate its use. The simulation tool is obviously not limited to the presented example applications, and here we discuss future applications of the tool, as well as its limitations. Studies over the last decades have shown that neurons in dLGN are not simple relays, and play much more important roles in information processing than previously appreciated [19,30,44,100]. (With this in mind, referring to dLGN neurons as ‘relay cells’ could be misleading, and a renaming could be in order. If so, a more neutral term to use could for instance be ‘thalamocortical cells’.) Compared to the primary visual cortex (V1), i. e. , the next station in the early visual pathway, the dLGN has received relatively little attention from computational neuroscientists [113]. From a modeling strategy point of view, this is somewhat unfortunate as progress towards a mechanistic understanding of the function of the dLGN circuit seems more attainable given that (i) the dLGN circuit involves much fewer neuron types and is more comprehensively mapped out [1,30], and that (ii) the dLGN has much fewer neurons making simulations computationally less intensive (18000 neurons in dLGN vs. 360000 neurons in V1 in mouse [114,115]). Further, the strong recurrent interactions characteristic for cortical networks (which make them difficult to understand and analyze) appear absent between the principal cells, that is, the relay cells, in the dLGN, even if key circuit network motifs such as feedforward and feedback interactions are present. Thus a focused and comprehensive effort on mechanistic modeling of the dLGN circuit would not only be of interest in itself, it would also likely be a very useful stepping stone for later attempts to model the visual cortex. While network simulations based on integrate-and-fire type neuron models (e. g. , [96,110]) and biophysically-detailed neuron models (e. g. , [38,97]) for entire dLGN nuclei are becoming computationally feasible with modern computers, there will still be a need for conceptually and mathematically simpler network simulation tools such as the present pyLGN tool based on the eDOG model. Such models are important to gain intuition about how the different circuit components may affect the overall circuit behavior, and will also be important for guiding the choice of the numerous, typically unknown, parameters in more comprehensive dLGN network simulations. Thus we envision that a future mathematics-based understanding of the dLGN circuit will be of a ‘multiscale’ nature and be based on a set of interconnected models at different levels of biophysical detail. The mouse seems particularly suitable as model animal since construction and testing of the multiscale models can be greatly facilitated by the ever more sophisticated techniques for controlling gene expression in mice as well as the possibility for optogenetic activation [114,116]. There are however differences in the way the mouse dLGN is organized compared to monkey and cat, which the current study has been based on, that should be considered [30]. For instance, although there are regions in the mouse dLGN clustered by morphology and ocular dominance, there are no clear laminations (except for a core and a dorsal shell region), and it has been difficult to identify functionally distinct classes of thalamocortical cells [117]. With respect to the model presented in this work, the main challenge is the lack of experimental data on whether the cortical feedback in rodents is organized in a phase-reversed arrangement (ON-ON feedback connections accompanied by OFF-ON connections) as observed in cats [73]. Experimental recordings to characterize this will be of particular interest since several computational studies have based their model on this organization, and in one study where the biphasic responses in LGN was explained using a predictive coding model, this arrangement of feedback emerged during training of the network [37,93,102]. We envision that at a close and targeted collaboration between modelers and experimentalist, in particular direct assessment of predictions from mechanistic models in targeted experiments, holds great promise for unraveling mechanisms of visual information processing at the different levels of advancements in visual information processing from the earliest sensory systems to more complex computations of the higher cortical areas.
On route from the retina to primary visual cortex, visually evoked signals have to pass through the dorsal lateral geniculate nucleus (dLGN). However, this is not an exclusive feedforward flow of information as feedback exists from neurons in the cortex back to both relay cells and interneurons in the dLGN. The functional role of this feedback remains mostly unresolved. Here, we use a firing-rate model, the extended difference-of-Gaussians (eDOG) model, to explore cortical feedback effects on visual responses of dLGN relay cells. Our analysis indicates that a particular mix of excitatory and inhibitory cortical feedback agrees best with available experimental observations. In this configuration ON-center relay cells receive both excitatory and (indirect) inhibitory feedback from ON-center cortical cells (ON-ON feedback) where the excitatory feedback is fast and spatially narrow while the inhibitory feedback is slow and spatially widespread. In addition to the ON-ON feedback, the connections are accompanied by OFF-ON connections following a so-called phase-reversed (push-pull) arrangement. To facilitate further applications of the model, we have made the Python tool pyLGN which allows for easy modification and evaluation of the a priori quite general eDOG model to new situations.
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Firing-rate based network modeling of the dLGN circuit: Effects of cortical feedback on spatiotemporal response properties of relay cells