Spatial and temporal coding in an olfaction-inspired network model Jeffrey Groff1, Corrie Camalier2, Cindy Chiu3, Ian Miller4 and Geraldine Wright5 1
2
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187; Center for Integrative and Cognitive 3 Neuroscience, Vanderbilt, Nashville,TN 37235; Department of Neuroscience, New York University, New York, NY 10011; 4 5 Department of Theoretical and Applied Mathmatics, University of Akron, Akron, OH 44325; Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210
Olfactory systems must solve a difficult combinatorial problem. Based on the architecture of the first synaptic relay of the olfactory system, we developed a model using Hodgkin-Huxley like equations to examine how local inhibition can contribute to developing a pattern of spiking activity that is related to features of synaptic input such as input strength. We also examine how changing synaptic strength, as may happen during learning, can influence the patterns of spiking activity of a network. Keywords: olfaction, Hodgkin-Huxley equations, odor coding, synchrony, network This paper is a brief description of a group project undertaken at the MBI during the Summer Education Program 2003 and as such includes general methods and analysis. For specific information regarding the modeling methods used in this project contact JR Groff, The College of William and Mary,
[email protected]. This funding to produce this report came from the National Science Foundation (Agreement No.: 0112050).
Introduction The olfactory systems of animals must code for 1,000’s of different volatile odorant compounds to perform tasks such as mate recognition and food identification essential to each organism’s survival. Most naturally occurring odors are combinations of many individual compounds making olfaction a complex sensory problem. Different phyla of animals have evolved to process odor stimuli in very similar ways (Hildebrand and Shepherd, 1997; Eisthen, 2002). This suggests that the morphology of the olfactory system plays a crucial role in odor perception and identification. Specific patterns of excitatory sensory input from sensory cells called olfactory receptor neurons (ORNs) appear to produce a combinatorial code for specific odorants (Malnic et al., 1999; Firestein, 2001). Several researchers believe that this sensory input is further organized and “coded” at the level of olfactory bulb (OB) (in vertebrate animals) or the antennal lobe (AL) or olfactory lobe (in invertebrate animals) (Laurent, 2002). From the OB or AL, the neural representation of odor is relayed to higher centers of the brain, such as the mushroom body (MB) in invertebrates, and the pre-piriform cortex in vertebrates where higher processing occurs. The specific mechanisms of odorant coding are still largely unknown. However, it is believed that both spatial and temporal patterns of active neurons in olfactory centers contribute to combinatorial codes for odorants (Malnic et al., 1999; Ma and Shepherd, 2000; Firestein, 2001). In locusts, electrophysiological recordings suggest that the code for odor identity arises as a result of synchronous spiking of projection neurons in the area of the AL that relays
information to the Kenyon cells in the MB in the locust brain (Laurent et al., 2001; Laurent, 2002). The purpose of this project was to construct a small network model to represent olfactory neurons in the AL of an insect in order to better understand how such a model behaves spatially and temporally. We also attempted to draw parallels between how our model responds to input and what is known about how real olfactory networks respond to odor stimuli. Specifically, we studied how modifications to the deterministic connectivity of neuronal compartments in our model affected patterns of response in our network. In this regard we looked at both excitatory and inhibitory connectivity. Additionally, we introduced a slowly activated, persistent membrane current representative of a metabotropic receptor G-protein coupled second messenger pathway. Through this mechanism we hoped to study how network plasticity or memory of recent odorants affected the neural code of future odorants either dissimilar or similar in nature to past odorants. General methods The morphology of the early olfactory system, the AL in insects, reflects its primary purpose, making discriminations between large numbers of different stimuli (Firestein 2001). ORNs of the main olfactory epithelium (MOE) send unbranching axons to specific spherical targets of neuropil known as glomeruli (Firestein 2001). Of functional importance is the fact that each ORN expresses a single receptor protein with a relatively specific chemical affinity, and all the ORNs expressing a particular protein project to one or a few specific glomeruli (Firestein 2001). Residing in the deeper external plexiform layer (EPL), mitral cells (MCs) send dendrites to one or a few glomeruli as well making specific synaptic contact with ORNs of similar receptor expression (Laurent 2002, Firestein 2001). MCs are the main projection neurons of early olfaction sending their axons to higher areas of the brain such as the periform cortex in mammals or the mushroom body in insects (Laurent 2002, Firestein 2001). Lateral processing occurs on the level of the glomeruli through periglomerular cells (PCs) and on the level of the MC’s by granule cells (GCs) (Firestein 2001). In insects, the complex behavior of the olfactory system is a product of the interactions between the different cell types and populations in the AL. It is thought that both local excitatory connections between ORNs and MCs and the lateral inhibitory connections provided by the lateral processing cell types described above contribute greatly to network behavior (Laurent 2002). For our purposes, we made the assumption that the synaptic characteristics, namely local excitation and a more distant lateral inhibition, contribute most significantly to olfactory network behavior. Thus, we constructed a simplified version of the complex multi-layer olfactory system by compressing all layers into a single onedimensional array of identical neuronal compartments. However, we conserved the overall synaptic properties of the network by imposing local excitation and a more distant lateral inhibition between compartments. We constructed our model in MatLab using reduced Hodgkin-Huxley type equations to model the intrinsic spiking currents. Each compartment had a fast
EPSP generating current consistent with L-glutamate mediated AMPA type receptors and a fast IPSP generating current consistent with GABAA receptor behavior. These synaptic gating variables were modeled using kinetic models as in equation 1 with different reversal potential establishing the difference between excitatory and inhibitory currents. I x = g x ⋅ S x ⋅ (V post − Vx ) (1) dS x = α x ⋅ (1 − S x ) ⋅ S∞ (V pre ) − β x ⋅ S x dt Synaptic spread was represented mathematically by a normalized synaptic footprint function with parameter λ. (equation 2) In order to model local excitatory and a more laterally distant inhibition we used a Gaussian footprint function for the AMPA-type current and a difference of Gaussian footprint function for the GABAA-type current consisting of an ‘off’ center and an ‘on’ surround. (Figure1)
(2)
− x2 f e = exp 2 2λ − x2 − x2 fi = exp − exp 2 2 ( 2λ )2 2λ
Figure 1. Example of normalized excitatory synaptic footprint (green) and inhibitory synaptic footprint (blue).
Additionally, each compartment possessed a second excitatory synaptic current governed by a Gaussian footprint. However, this current activated slowly upon repeated excitation by EPSPs. Initially, contributions from this synaptic
current are weak. However, upon repeated excitation of the compartment via IAMPA, this current activates and contributes more significantly upon future spiking events. In addition, this current deactivates on a much slower time scale on the order of minutes. Thus, its effects can be felt far into the future. Physiologically, this additional excitatory current can be thought of as the phenomenon of synaptic strengthening due to repetitive excitatory input. In the framework of this project, this synaptic strengthening can be thought of as a learning mechanism. The means through which recruitment of this current happens is analogous to a metabotropic receptor second messenger cascade. We modeled this mechanism using the equations found in equation 3. I S = g S ⋅ S S ⋅ PM ⋅ (V post − VS ) dS S = α S ⋅ (1 − S S ) ⋅ S∞ (V pre ) − β S ⋅ S S dt (3) dPM = α M ⋅ M ⋅ (1 − S M ) − β M ⋅ S M dt dM = −ν ⋅ I AMPA − γ ⋅ M dt
In the above equations, IS represents the excitatory synaptic current attributed to synaptic strengthening, SS is a synaptic gating variable activating on the order of time relative to a single action potential, PM is the synaptic gating variable activated by second messenger, M, concentration. M concentration is a function of the AMPA-type excitatory synaptic current, IAMPA The development of temporal synchrony as a function of gAMPA and gS as applies to our olfactory inspired network It is known that excitatory synaptic connectivity plays a role in synchronizing the behavior of neural networks. Our model has two such sources of excitatory synaptic conductivity, namely the AMPA-type current, IAMPA, and the current due to synaptic strengthening described above, IS. Thus, if the network is initialized with asynchronous initial conditions and given excitatory full-field stimulation through a constant applied current, one would expect the network to synchronize its spiking behavior at some future time.
Figure 2. Network synchrony in an array of cells (x-axis) as a function of time (y-axis) with cell voltage in and out of the board.
Notice in the equations above (equation 3) that the rate at which the second messenger concentration grows, and thus ultimately the increase in the strength of IS itself, is a function of IAMPA. Thus, the strength of IS as a function of time and thus the speed at which the network synchronizes its behavior is a function of gAMPA, the maximum conductance of the AMPA-type synaptic current, and gS, the maximum conductance of the slowly activated excitatory synaptic current. By systematically changing gAMPA and gS and subjectively observing the degree of synchrony that our model was able to achieve in a given time interval, we were able to make some general observations regarding the interplay of gAMPA and gS. These observations are briefly outlined below. 1) As gAMPA increases, IAMPA increases, thus, the rate at which IS activates increases and the rate at which the network exhibits synchrony increases 2) As gS increases, IS increases for a given level of activation, thus the rate at which the network becomes synchronous increases 3) There is an inverse relation between values of gAMPA and gS that can be used to achieve a given level of synchrony in a given time interval. 4) There is an optimal value for gAMPA and gS that achieves a given level of synchrony in a given time interval. By this we imply that the sum of gAMPA and gS is minimized.
Figure 3. Illustration of the qualitative relation that exists between gAMPA and gS values that result in a given level of synchrony in a given period of time.
gAMPA vs. gS
From a biological prospective, observation 4 becomes very enlightening. “g” values can be thought of biologically as the surface density of ion channels in the cell membrane. As most scientists know, protein synthesis and maintenance is a metabolically “expensive” task. Thus, a cell will want to maximize the usefulness of the proteins it has by achieving a desired effect with a minimum number of proteins. Our analysis shows that if the desired effect is known, in this case network synchrony, and the contributing factors are understood, g values for example, then the optimal values for the contributing factors can be derived computationally. I. An investigation of the effects of inhibitory and excitatory synaptic footprint size and relative stimulus strength on the sensitivity and resolution of our olfactory inspired network. Several studies have shown that ORNs expressing the same receptor protein converge to the same glomerulus in the AL or olfactory bulb. These receptor proteins tend to bind preferentially with specific features of odour molecules. It has been hypothesized, therefore, that the combinations of responses of ORN cells to specific features of odour molecules may provide a combinatorial code for odour identity that is represented in the spatial activity in the antennal lobe or olfactory bulb. Odours that have similar chemical components would be expected to activate overlapping populations of ORNs, and therefore would also activate patterns in the glomeruli of the antennal lobe/mushroom body (AL/OB) that would overlap. In mixtures of odours, one would also predict that mixtures containing dissimilar odours would produce patterns of activity in the AL/OB that were
spatially distinct and mixtures of similar smelling odours would be highly overlapping. We propose that the ability of an olfactory network to discern between similar odorants, i.e. spatial resolution, is a function of odorant strength and the size (or reach) and strength of excitatory and inhibitory synaptic connectivity. We tested this using our representative olfactory network. Stimuli to our network took the form of Gaussian distributed applied currents that could be centred on an arbitrary position of our one-dimensional array of cells and could have an arbitrary width. In light of the discussion above, we simulated similarity between stimuli (odorants) by spatial proximity. Two stimuli centred far away from one another on our network were assumed to correspond to a response indicative of chemically different odours. Concomitantly, stimuli that were proximal or even overlapping were analogous to olfactory stimulation by chemically similar odorants. For these experiments the learning parameter of our model, measured as Gadap, is deactivated. Network behavior therefore reflects transient activity over a population of ten neurons, with a low time constant. We consider three cases of spatial input. In the first case, the distance of the two inputs is large enough that no overlap of the local excitation occurs. In the second case the distance between the two inputs is reduced allowing for mild overlap in the local excitation between the competing stimuli. In the last case, there is high overlap between competing stimuli. In all three cases, Ggaba is varied along the values of {0, 1, 3, 5} and Gampa is varied along {0, 1, 1.5, 2, 2.5, 3, 5}. The full dataset is produced from the Cartesian product of the sets of Gampa and Ggaba parameters. For the second experiment, only the case with the highest level of overlap between input stimuli is considered to determine discriminability of highly similar odor percepts. Analyzing the cases of optimal Ggaba coupling (values of 3 and 5), we modulate the strengths of the input signals to 0.5, 1.5, 2, 3 times the strength of the stimulation of the original signal (10). In the first set of simulations, examining the roles of Gampa and Ggaba, a number of trends emerged over the range examined. Over all, ranges of stimulus input, higher values of Gampa (inhibitory coupling strength) corresponded to more coordinated recruitment of the network by an individual stimulus. If sufficiently high, the simultaneous firing from both stimuli overlapped and could not be discriminated. Higher values of Ggaba (excitatory coupling strength) revealed a more localized firing response in the neighbourhood of the input and the emergence of stronger oscillations between firing of the network regions. Due to these behaviours, discrimination of the distinct signals was possible for the three input distances considered, dependent upon a sufficiently high Ggaba, in the range of 4 or 5, and a bounded Gampa, in the range of 1 to 3. The second set of simulations focussed on the effect of signal strength on discriminability. We observed a positive correlation between signal strength and discriminatory ability. As signal strength increased, we observed greater network recruitment over time and more pronounced oscillatory behaviour between firing regions. Also, the frequency of oscillations is reduced with greater signal strength, so activated regions stay activated for longer periods of time. In
general, signal strength was correlated with a discretized response localized both temporally and spatially over the network response. An olfactory system representation must have the ability to differentially represent the identities of an odour mixture. A logical assumption would be that similar odours will activate partially overlapping combinations of receptors with differing strengths of activation. Here, the model seems to suggest that for any two signals to remain discriminable, the network must have sufficiently high local excitation to propagate the signal, and an optimal level of inhibition to focus the signal to its region on the network. The network tuned in this way to produce a discretized signal over the parameters chosen invariably results in temporal oscillations between the network responses for either stimulus. This oscillatory activity can probably be attributed to post inhibitory rebound between these inhibition-coupled stimuli, and reflects another mechanism for the separation of two concurrent signals. Although certainly compelling, it is important to emphasize the preliminary nature of these data, as many aspects of the model have yet to be fully explored, such as the effect of different magnitudes of noise on the system. There are many factors in the model that affect network behaviour, and these trends observed can serve as a preliminary behavioural exploration into the interaction between synaptic architecture and the position and magnitude of a given odour representation in the antennal lobe or olfactory bulb. II. Investigation of Stimuli Learning The olfactory system must code for 1,000’s of odors that are comprised of 1,000’s of different odorant molecules. Instead of being a “labeled line” code for odor identity, it is likely that the plasticity of the olfactory system produces a code for odor identity that animals use to recognize previously experienced odorants (Laurent et al., 2001; Laurent, 2002). One possible mechanism underlying the function of the olfactory system may be the ability of the neurons in the AL/OB to “learn” to coordinate their activity as a result of sensory input (Hudson, 1999). Evidence for increasing coordination between neurons of the AL/OB as a result of experience comes from electrophysiological studies of the antennal lobe of locusts (Laurent and Davidowitz, 1994; Laurent et al., 1996; Stopfer and Laurent, 1999). These studies have shown that repeated presentations of odor result in an increase in the coordination of the spike timing, or synchronization, of the responses of projection neurons in the AL (Stopfer and Laurent, 1999). Upon each presentation of odor, the firing activity decreases and becomes more synchronous among projection neurons. Our network already has a mechanism for learning in the form of IS. Thus, we conducted an experiment to see if “learning” a smell, i.e. repeated applications of the same stimulus, resulted in increased network synchrony at subsequent presentations of the stimulus. However, we considered the unique consequences of learning 2 similar stimuli simultaneously. Our criterion for “learning” was the level of activation of the excitatory synaptic current due to synaptic strengthening, PM in equation 3.
Our stimuli took the form of applied currents upon the one-dimensional array of neuron compartments. The applied current was distributed as a Gaussian so that it could be centered on a given compartment and would go to zero at the periphery. In our computations, the idea of chemical similarity between odors was simulated by the spatial proximity of stimuli on the onedimensional array of neuron compartments. Closer spatial proximity was analogous to closer chemical resemblance. This assumption is correct if the chemoreceptor for smell are spatially arranged by preferred chemical affinity. We found that if two stimuli were significantly similar, i.e. spatially localized on the network, then the learning phenomenon of overshadowing was observed. Overshadowing is when two similar stimuli, A and B, are presented simultaneously but neither stimuli is learned as effectively as if A and B were presented independently. Figure 4. The learning mechanism of overshadowing and blocking (not observed in our investigation. Each arrow indicates the input stimuli.
Recall that our model possesses a local excitatory synaptic component but also a more distant inhibitory synaptic component. We propose that overshadowing is the result of the overlap of the inhibitory footprint of one stimulus with the excitatory footprint of an adjacent stimulus. In this configuration the neighboring populations of excited neurons resulting from stimulation will be inhibiting each other, thus neither population will learn the stimulus presented to it as well as if that was the only stimulus the network was “seeing”.
Figure 5. An illustration of how two spatially localized stimuli (right) can have overlapping inhibitory and excitatory footprints thus interfering with the networks ability to learn. Each arrow indicates where input arrived simultaneously.
General conclusions and Future Directions We have shown that our network of neurons can discriminate among inputs impinging from different locations and that it can also learn to recognize specific inputs. We have also shown that the ability of our network to perform these functions is dependent upon the excitatory strength between neurons and the strength and extent of inhibition among neurons in our network. The strength of the input also influenced the behavior of the network. We have several hypotheses that could be tested to follow up on our preliminary results: 1. Follow up on the ability of the network to discriminate odors. A description of the extent to which overlapping odors can be discriminated would benefit from a more detailed analysis. The current model already has the capability to do this as overlapping odors may be represented by placing the centers of the stimuli at close range and using a broader footprint for the applied current. We would examine in further detail the temporal structure of the response of overlapping sets of cells to see how the parameters of our model might influence this pattern of activity to code for “odor” identity. 2. Simulate an experimental paradigm where odors are presented in puffs. By adding applied current to the network in brief pulses with longer periods of delay, we predict that the network would be able to learn an odor over time. This might be realized by a decreasing time to oscillatory behavior after each successive application of current, as was observed by Stopfer and Laurent (1999). 3. Rework network architecture to allow for more than one population of cells. Instead of using a one-dimensional array of N cells, we might build a population where the inhibitory connections are actually made via a nonlinear node providing a negative feedback loop. In this situation, inhibitory connections can be also be made between the populations of cells, which is more similar with the biological architecture of the olfactory system. Our current 1-D model is similar to the case of Bazhenov et. al.’s (2001) simple 2D network model of 6 pn’s and 2 ln’s in which the ln-ln connections were blocked. In this case, they observed sustained synchronous activity, which
was also our observation. As suggested by their model, a more realistic transient synchrony can only be observed when inhibitory connections between two populations are implemented. Acknowledgements: We would like to thank Michael Smith, and Matthew Wallschlaeger for help during the project, and David Terman for his insightful input. This work was supported by the National Science Foundation (Agreement No. 0112050). References: Bazhenov, M., Stopfer, M., Rabinovich, M., Huerta, R., Abarbanel, H.R.I., Sejinowski,T., and Laurent, G. 2001. Model of transient oscillatory synchrony in the locust antennal lobe. Neuron 30: 553-567. Eisthen H.L. 2002. Why are olfactory systems of different animals so similar? Brain Behavior and Evolution 59 (5-6): 273-293 Firestein, S. 2001. How the olfactory system makes sense of scents. Nature 413, 211-218. Hildebrand, J.G. and Shepherd, G.M. 1997. Mechanisms of olfactory discrimination: converging evidence for common principles across phyla. Annu. Rev. Neurosci. 20: 595-631. Hudson, R. 1999. From molecule to mind: the role of experience in shaping olfactory function. J. Comp. Phys A 185: 297-304. Laurent, G. 2002 Olfactory network dynamics and the coding of multidimensional signals. Nat. Rev. Neurosci. 3 (11): 884-895. Laurent, G. and Davidowitz, H. 1994. Encoding of olfactory information with oscillating neural assemblies. Science 265 (5180): 1872-1875. Laurent, G., Wehr, M. and Davidowitz, H. 1996. Temporal representation of odors in an olfactory network. J. Neurosci. 16 (12): 3837-3847. Laurent, G., Stopfer, M., Friedrich, R.W., Rabinovich, M.I., Volkovskii, A., and Abarbanel, H.D.I 2001. Odor encoding as an active, dynamical process: experiments, computation, and theory. Ann. Rev. Neurosci. 24, 263-297. Ma, M.H. and Shepherd, G.M. 2000. Functional mosaic organization of mouse olfactory receptor neurons. Proc. Natl. Acad. Sci USA 97, 12869-12874. Malnic B, Hirono J, Sato T, Buck LB. 1999. Combinatorial receptor codes for odors. Cell 96 (5): 713-723. Stopfer, M. and Laurent, G. 1999. Short-term memory in olfactory network dynamics. Nature 402 (6762): 664-668.
