The piriform cortex is rich in recurrent excitatory synaptic connections between pyramidal neurons. We asked how such connections could shape cortical responses to olfactory lateral olfactory tract (LOT) inputs. For this, we constructed a computational network model of anterior piriform cortex with 2000 multicompartment, multiconductance neurons (500 semilunar, 1000 layer 2 and 500 layer 3 pyramids; 200 superficial interneurons of two types; 500 deep interneurons of three types; 500 LOT afferents), incorporating published and unpublished data. With a given distribution of LOT firing patterns, and increasing the strength of recurrent excitation, a small number of firing patterns were observed in pyramidal cell networks: first, sparse firings; then temporally and spatially concentrated epochs of action potentials, wherein each neuron fires one or two spikes; then more synchronized events, associated with bursts of action potentials in some pyramidal neurons. We suggest that one function of anterior piriform cortex is to transform ongoing streams of input spikes into temporally focused spike patterns, called here “cell assemblies”, that are salient for downstream projection areas.
The circuitry of the piriform cortex has been reviewed extensively, both in terms of cell and also synaptic/circuit properties (Bekkers and Suzuki 2013; Haberly 2001; Poo and Isaacson 2009, 2011; Stern et al. 2018; Stokes and Isaacson 2010; Suzuki and Bekkers 2006, 2007, 2010a,b, 2011). There are three main types of principal neuron: semilunar cells (Choy et al. 2017; Suzuki and Bekkers 2006, 2011), and layer 2 and layer 3 pyramidal cells (here designated L2pyr and L3pyr, respectively). While semilunar cells act primarily in feedforward fashion with few recurrents from any of the principal neuron types (Choy et al. 2017; Suzuki and Bekkers 2006, 2011), the pyramidal cells are extensively interconnected by excitatory collaterals (Franks et al. 2011). These collaterals are spatially distributed, as are the afferent inputs from the lateral olfactory tract (LOT), so that there is little evidence for either a topographic or columnar organization (reviewed by Srinivasan and Stevens (2018)).
The prominent recurrent circuitry led Haberly (2001) to propose that olfactory cortex stores correlations between “olfactory features” characterizing various odors. This idea has been expanded upon, in terms of the plastic properties of recurrent synapses, by Hasselmo, Barkai, and others (Barkai et al. 1994; Barkai and Hasselmo 1997; Katori et al. 2018; Saar et al. 2001). Fournier et al. (2015) have proposed that the three-layered cortices, such as olfactory cortex (and hippocampus), may be suitable for elucidation of general computational principles, providing further impetus to model this structure.
Dense excitatory synaptic connectivity also suggests the possibility that so-called “cell assemblies” could be generated. The term “cell assembly” is attributed to Donald Hebb (1949); it means different things to different investigators (Buzsáki 2010; Garagnani et al. 2009; Harris 2005; Holtmaat and Caroni 2016; Huyck and Passmore 2013; Kopell et al. 2010; Palm et al. 2014; Reichinnek et al. 2010; Sakurai 1999; Traub et al. 2000). Some examples of overlapping, yet distinct, meanings of the term are these: a set of neurons that are recruited to fire as a consequence of stimulation of a subset of them, as in an associative memory; a set excitatory neurons, interconnected synaptically, that exhibits reverberating activity, either spontaneously or in response to local stimulation; a set of neurons that fire action potentials synchronously, on a time scale defined by the investigator.
That cell assemblies form in piriform cortex, in a manner that is relevant to observable behavior, is suggested by experiments of Axel and collaborators, in which optogenetic stimulation of subsets of olfactory cortex pyramidal neurons could lead to one or another distinct behavior (Choi et al. 2011); the optogenetic stimulation in question lasted for seconds, a time scale that may be comparable to what occurs during spontaneous olfaction-guided behavior in vivo (Stevens et al. 1988).
The main issues we seek to address are these: How do patterns of pyramidal cell activity emerge in a network with the known features of anterior piriform cortex, and should such patterns be designated cell assemblies? What are some of the key parameters that influence such patterns? Are the principles involved similar to those involved in generating synchrony within the CA3 region of the hippocampus, another three-layered structure (Miles and Wong 1987; Traub and Miles 1991)? We address these questions primarily with a detailed computational model, in the spirit of Traub et al. (2005) and also with in vitro experimental recordings.
The network model consisted of 2700 multicompartment (∼100 compartments each), multiconductance neurons, plus 500 afferent axons. There were 500 semilunar cells; 1000 layer 2 and 500 layer 3 pyramids; 200 superficial interneurons of two types: “Type 1” (100) and neurogliaform (100); 500 deep interneurons (fast-spiking basket cells (200); neurogliaform (200); low-threshold spiking, dendrite-contacting interneurons that induced GABAA conductances with slower kinetics than the other interneurons (100) (Kapur et al. 1997, but see Large et al. 2016 for a somewhat contrasting view)). The division of interneurons into superficial and deep classes follows a number of studies (Stokes and Isaacson 2010; Suzuki and Bekkers 2007, 2010). We did not include axo-axonic interneurons (Larriva-Sahd 2010) or deep multipolar cells (either excitatory or inhibitory). Axo-axonic GABAergic synapses appear to be rare in piriform cortex (Ekstrand et al. 2001). There were also 500 LOT afferent fibers. The data base included over 450 network simulations, after preliminary test runs.
It is important to recognize the small size of this model (for reasons of computer tractability) relative to the biological structure (see Discussion). For example, Srinivasan and Stevens (2018) estimate that there are about 500,000 L2pyr and L3pyr in a mouse piriform cortex (anterior together with posterior). This point is enlarged upon in the Discussion. We note, however, that the model cannot be too small as that would make it impossible to observe phase transition-like behavior (as in Figure 7 below), or to understand the transition from population single/double spikes to population bursts. Our experience over the years suggests that models with some thousands of cells are appropriate for studying such phenomena (Traub and Miles 1991). A sufficiently large model (although it is hard to be precise here) also allows one to observe cell assemblies having a range of sizes relative to the entire population.
All the model neurons were multicompartment, with structure as indicated in Figure 1. The general computational principles were as in neocortical/thalamic models of Traub et al. (2005) and Carracedo et al. (2013) (for the neurogliaform cells). We note that the kinetic properties of many of the ionic conductances in piriform cortex pyramidal neurons have been reported to be similar to those in neocortical neurons (Ikeda et al. 2018). Each model cell had a soma, a short piece of axon, and branching dendrites. There were 74 compartments for the pyramidal cells, 50 for semilunar cells, and 59 for each of the interneurons. Semilunar cells were constructed from L2pyr by disconnecting the basal dendritic compartments, and L3pyr were constructed from L2pyr by elongating the apical dendrite. Full details are embedded in the Fortran code, available at ModelDB. Parameter values were determined by the literature, to the extent feasible, and by a good deal of trial and error. The available data are not sufficient to fully constrain all the parameters, and yet good qualitative agreement with experiment is often possible, both for current injection responses and for collective behaviors (c.f. Figure 4).
Passive parameters included membrane capacitance C m of 0.9 µF/cm2 for principal cells and 1.0 µF/cm2 for interneurons. Internal resistivity R i was 250 Ω-cm for principal soma/dendrites, 200 Ω-cm for interneuron soma/dendrites, and 100 Ω-cm for all axons. Membrane resistivity R m was 50,000 Ω-cm2 for principal soma/dendrites, 25,000 Ω-cm2 for interneuron soma/dendrites, and 1000 Ω-cm2 for all axons.
Voltage- and calcium-dependent ionic conductances that were simulated included the following: Fast sodium, gNa(F); persistent sodium gNa(P); high-threshold calcium gCa(L); low-threshold calcium gCa(T) (important in “LTS” interneurons); delayed rectifier gK(DR); A-type potassium gK(A); C-type voltage- and calcium-dependent potassium conductance with fast kinetics gK(C); slow afterhyperpolarization that is calcium-dependent gK(AHP) (and that was given faster kinetics than in neocortical simulations (Traub et al. 2005)); the slow gK2; and an anomalous rectifier or h-conductance gAR. The densities were dependent on the cell type and the compartment, with axons having typically higher gNa(F) density and faster kinetics than elsewhere, and principal cell dendrites having higher gCa(L) densities than elsewhere. The conductance kinetics were simulated with Hodgkin–Huxley-like formalisms.
The equilibrium potentials for the ionic conductances included the following: for the leak current, VL, −70 mV for principal neurons and −65 mV for interneurons; for K+ currents, VK, −95 mV for principal neurons and −100 mV for interneurons; VNa = +50 mV; VCa = +125 mV; VAR = −35 mV for principal neurons and −40 mV for interneurons; VGABA(A) = −81 mV for principal neurons and −75 mV for interneurons; VGlu (for AMPA- and NMDA-receptor mediated currents) = 0 mV.
Each unitary synaptic conductance consisted of a scaling term (e.g. gAMPA L2pyr/L2pyr for AMPA receptor mediated L2pyr→L2pyr connections) multiplied by a term that was time-dependent, and – in the case of NMDA conductances – also voltage- and [Mg2+]o-dependent. AMPA-receptor mediated conductances were initiated by presynaptic action potentials and were simulated as a scaling constant times an α-function, t × exp(−t/τ), where t = time in ms and τ = 2 ms for conductances on principal neurons, and 0.8 ms for connections on interneurons (Note that the α-function will achieve its maximum value when t = τ and this value will be τ/e ∼ τ/2.718). GABAA conductances also had a scaling constant, and rose instantaneously after a presynaptic action potential, and decayed with time constant 6 ms for principal cells (except 20 ms for inputs from “Type 1” superficial interneurons), and 3 ms for synapses on interneurons (Bartos et al. 2001). The voltage-dependence of NMDA conductances used the same formalism as before (Carracedo et al. 2013; Traub et al. 1994, 2005) with relaxation time constant 130.5 ms for synapses on principal neurons and 100 ms for synapses on interneurons. GABAB currents were also simulated using a previous formalism (Carracedo et al. 2013), with equilibrium potential = VK, but with a relatively fast decay time constant (100 ms); GABAB currents were produced by firing in superficial and deep neurogliaform cells. We did not simulate such currents in interneurons, and indeed in the present case these currents were small and had little influence on the results. The model did not include kainate receptors, although these are likely important in piriform cortex (Chandra et al. 2019). We also did not include presynaptic GABAB receptors (Gerrard et al. 2018). Synaptic plasticity was not simulated.
Details of layer 2 pyramidal cells (L2pyr)
We used the model of neocortical superficial pyramidal cells from earlier publications, integrate_suppyrRSXPB.f, a descendant of integrate_suppyrRS.f (Traub et al. 2005). Figure 2 illustrates the firing properties of the model cells, during somatic current injection, under different conditions. Note that bursting in the model cells is favored by reduced dendritic inhibition; simulations below (e.g. Figure 7) also demonstrate that the model cells can exhibit bursts during sufficiently strong synaptic excitation.
Details of semilunar cells
We used the above program for simulating L2pyr, but without basal dendrites, leaving 50 compartments in all.
Details of layer 3 pyramidal cells (L3pyr)
We used the integration program for L2pyr, with the same compartmental topology, but lengthening 3 of the compartments along the apical shaft from 50 to 65 µm.
Details of superficial “Type 1” interneurons
We used the code for superficial VIP interneurons (integrate_supVIP.f), which was in turn derived from the code for LTS interneurons (see below) (This is not meant to imply that the interneurons being so simulated are necessarily VIP-positive – there was simply a naming convention for the subroutine).
Details of superficial and deep neurogliaform cells
These were fast-spiking interneurons with increased gK(A) to render them “delayed spiking”, that is the first action potential in response to a depolarizing current pulse was delayed (Carracedo et al. 2013).
Details of deep fast-spiking “basket” cells
Details of deep low-threshold-spiking (“LTS”) interneurons
Synaptic connectivity densities and contact sites
We attempted to follow some of the basic anatomical data, while allowing for the relatively small scale of the model: for example, LOT afferents contact distal apical dendrites of principal cells and superficial interneurons (Figueres-Oñate et al. 2014; Stokes and Isaacson 2010); there is a division of interneurons into superficial and deep populations (Suzuki and Bekkers 2007, 2010a,b, 2011); the passage of association fibers in layer Ib, although not exclusively in that location (Luskin and Price 1983); there is little recurrent excitation of semilunar cells, either by other semilunar cells or by pyramidal cells (Choy et al. 2017).
We did not consider rostral/caudal inhomogeneities in association fiber connectivities (Haberly and Price 1978), given the relatively small size of the model network.
Some of the connection parameters are as follows (further details are in the code, in arrays such as “compallow”): 20 LOT fibers go to each L2pyr, 30 to each L3pyr, 30 to each semilunar – all to distal dendrites. Thirty LOT fibers go to each superficial Type 1 and each superficial neurogliaform interneuron, across multiple compartments. LOT fibers do not connect to deep interneurons. L2pyr receive input from 20 other L2pyr, 40 semilunar and 10 L3pyr, all scattered across dendrites except distal apical compartments; and from 20 of each of the five types of interneuron with superficial interneurons inhibiting distal dendrites and basket cells the soma and proximal dendrites. Semilunar cells receive input from only two of each type of principal cell; L3pyr from 10 L2pyr, 40 semilunar, 10 L3pyr. The interneuron connectivity to SL and L3pyr are as for L2pyr. Superficial interneurons receive weak excitation from 10 L2pyr each and 0, 1 or 5 L3pyr/semilunar cells. All three types of deep interneuron receive input from 60 L2pyr, 20 semilunar, and 40 L3pyr. Connectivity amongst interneurons is sparse except that 10 basket cells connect to each deep interneuron.
Synaptic kinetics and scaling
As described above, unitary synaptic conductances have the form
Scaling constant × term that is time-dependent (for NMDA also voltage- and [Mg2+]o-dependent).
The scaling constant gAMPA L2pyr/L2pyr was varied across many of the simulations, as noted in the figures and text. Other scaling constants were kept fixed, except as noted in the text. Some of the values were these: gAMPA LOT/L2pyr 15 nS, gAMPA LOT/L3pyr 15 nS, gAMPA LOT/SL 10 nS, gAMPA LOT/Type 1 4.5 nS, gAMPA LOT/supng 4.5 nS; gGABA(A) bask/L2pyr 2 nS, gGABA(A) bask/L3pyr 2 nS, gGABA(A) bask/SL 1 nS; gGABA Type 1/L2pyr 1.75 nS, gGABA Type 1/L3pyr 1.75 nS, and gGABA Type 1/SL 0.5 nS. Other values are in the code.
Although it is known that short-term plasticity occurs at various synapses in the piriform circuitry (Suzuki and Bekkers 2006), for the sake of simplicity we did not simulate these effects.
For comparison with experimental data, a single fiber LOT/L2pyr EPSC can reach >100 pA, although some of these EPSCs are smaller (Franks and Isaacson 2006). These authors found that an average of about 300 pA EPSC was required to reach spike threshold in a pyramidal cell, or roughly three LOT inputs (assuming simultaneity). For one comparison with our model (where principal cells have resting potential −70 mV), in simulation “piriformECT10”, gAMPA LOT/L2pyr was 15 nS, giving an estimated unitary EPSC of about 750 pA (and a similar value for L3pyr). For LOT to SL, an estimated model unitary EPSC was about 500 pA.
Gap junctions (interneuron, principal cell)
Simulated gap junctions were only allowed between homologous neuron types. Electrical coupling between the axons of principal cells was kept, in almost all simulations, below the “percolation limit” (see Traub et al. (1999)), and the coupling conductance small enough to make unlikely action potential crossings from axon to axon (e.g. 5 nS for L2pyr, 3 nS for semilunar and L3pyr). There was electrical coupling between interneurons, except for LTS cells: 250 junctions (conductance 1 nS) in the basket cell population; 1000 junctions (2 nS) in the superficial FS population; 250 junctions (0.5 nS) in each of the neurogliaform populations.
Driving currents, pseudorandom spike trains
Small tonic bias currents were delivered to the somata of the model neurons, and to the dendrites of principal neurons, with the purpose of introducing heterogeneity into cellular intrinsic properties; however, we did not attempt to simulate the spontaneous activity of pyramidal neurons that has been described in vivo (Tantirigama et al. 2017). It was noted that interneurons were sensitive to small bias currents, particularly basket cells, so that some simulations were repeated with different interneuron bias. In most cases, the bias to interneuron somata were these: For basket cells −0.1 to −0.08 nA, to suppress spontaneous firing; for main superficial interneurons, 0.13–0.14 nA; for superficial neurogliaform, −0.03 nA; for deep neurogliaform, −0.045 nA. Deep LTS interneurons were not biased – an arbitrary choice and one highlighting a need for more experimental data – rendering them sensitive to pyramidal cell activity. In a few simulations, a pseudorandom background of ectopic axonal spikes was present in the principal neurons; these had Poisson statistics, as in previous publications (Traub et al. 1999).
Qualitative nature of the simulations
We used the model to examine qualitative aspects of network behavior rather than quantitative (for example, the shape of the curve in Figure 7 rather than the exact synaptic conductance at which the inflection occurs, or the precise amount of axonal activity for any particular parameter choice). This approach was further motivated by the small size of the model, as we examine in detail in the Discussion. The qualitative approach is in the spirit of other modeling endeavors including (a) the use of simplified models (e.g. Fitzhugh–Nagumo (Fitzhugh 1961)) of action potential generation to highlight qualitative features, and (b) models of phase transitions in statistical physics in which important aspects of system behavior around the transition are predicted, but no quantitative prediction of the transition temperature is possible (An example of the latter is the Ising model of ferromagnetism (Onsager 1944)).
Testing for robustness of the simulations
Due to the use of either constant or pseudorandom afferent inputs, and to the deterministic nature of the ongoing simulation (i.e. synaptic transmission in the model is not stochastic, and structural parameters are time-invariant), each simulation is deterministic from a computer science point of view: that is, if a simulation is repeated, it generates exactly the same results as the first time. One could, of course, repeat simulations with a different seed for the pseudorandom inputs. In order to be sure of robustness of the results, we took another approach that may be more biologically meaningful: that is, we demonstrated robustness by repeating series of simulations (such as those in Figure 7) with somewhat different parameter choices (e.g. different patterns of afferent stimulation), or blocking selected synaptic conductances other than the primary one being investigated (typically gAMPA L2pyr/L2pyr). These additional simulations are not illustrated here, but gave results similar to those we do illustrate.
Comparing simulations with experimental multiunit recordings
With a simulation such as described here, the programmer has access to all structural parameters and spontaneous activities. It is therefore possible, in principle, to account for every action potential in every cell, with (so to speak) a history. This is one of the uses to which the model will eventually be put: that is, if certain pyramidal cells are induced to fire by afferents, one can analyze which other cells will fire and when, and under what conditions. In a typical multiunit experiment, however, although the experimenter can determine firing times of various selected neurons, the underlying structural details are inaccessible to observation. As a consequence, then, if the selected neurons appear to form an assembly (that is to be synchronized), it is necessary to use a statistical analysis to determine if the synchrony occurred by chance (according to some criterion). In our model, however, this type of chance is not a factor. This consideration is vital to understanding our simulation data.
Local field potentials
We did not attempt to estimate these explicitly, but as a substitute used the inverted average somatic potential of 500 L2pyr.
Code was written in Fortran and is available from email@example.com. It uses the mpi parallel environment. Differential equations were integrated with a second order Taylor series method with time step 2 µs.
The program typically ran on an IBM Power GPU compute node residing in the IBM Cognitive Computing Cluster and running Linux. Top-level program names include: piriform.f, piriformA.f, piriformGJ.f, piriformECT.f and piriformPERC.f. A simulation of 1.2 s of network activity ran for approximately 3.2 h (Code has been deposited at the Yale NEURON website, senselab.med.yale.edu [ModelDB], accession 266902. The files there include the makefile and run command, as well as a sample pdf of output).
Coronal sections, 450 μm thick, containing anterior piriform cortex were prepared from brains of adult male C57b6 mice (20–25 g) following cardiac perfusion with ice-cold buffered sucrose acsf (artificial cerebrospinal fluid) (252 mM sucrose, 3 mM KCl, 1.25 mM NaH2PO4, 24 mM NaHCO3, 2 mM MgSO4, 2 mM CaCl2, 10 mM glucose). Slices were maintained at 33 °C at the interface between acsf (126 mM NaCl, 3 mM KCl, 1.25 mM NaH2PO4, 24 mM NaHCO3, 1 mM MgSO4, 1.2 mM CaCl2, 10 mM glucose) and warm, wetted 95% O2/5% CO2. All surgical procedures were performed in accordance with the UK Animals (Scientific Procedures) Act, 1986 and the University of York animal welfare committee.
Local field potentials were recorded from coronal sections using sharp glass microelectrodes (resistance >1 MΩ) filled with acsf; intracellular recordings of presumed layer 2 pyramidal cells used sharp electrodes (50–90 MΩ) filled with 2 M potassium acetate. Pyramidal cells were delineated from semilunar cells on the basis of Suzuki and Bekkers (2011) – the latter spike late on current injection, have no initial spike doublet and do not overtly accommodate. All slices were quiescent without pharmacological manipulation. Bath application of carbachol (2–5 μM) was used to generate persistent spontaneous activity, with this concentration range producing a marked increase in EPSP frequency (no-drug vs. drug, 7.2 ± 0.3 vs. 27 ± 0.4 s−1, P < 0.05, 1650 events from n = 4 cells (no drug) and 8760 events from n = 4 cells (carbachol)). A small but significant decrease in mean EPSP amplitude was also seen (no-drug vs. drug, 1.1 ± 0.2 vs. 0.8 ± 0.2 mV, P < 0.05 at a mean membrane potential of −67 mV and input resistance of 51 MΩ).
For comparison, Wiegand et al. (2011) estimate L2pyr input resistance, with patch recordings, at about 200 MΩ. Their data would give an EPSC of about 5 pA, perhaps more allowing for the dendritic site of synaptic input. For our cells and input resistance about 50 MΩ, the recurrent EPSC is estimated at about 20 pA. Interestingly, Figure 4C of Wiegand et al. (2011) indicates a correlation between recurrent connectivity and the tendency of pyramidal neurons to burst (Conditions under which model neurons burst will be considered below).
To examine the behavior of the model network, we used several stimulation paradigms, including current pulses injected into a single pyramidal cells; Poisson-distributed (in time) action potentials in the LOT, either in all of the axons or in various defined subsets, and at various mean rates; periodic current pulses to subsets of LOT (Poo and Isaacson 2009), and steady currents to LOT fibers (Actual mitral/tufted cell axons fire in quite complex temporal patterns (Balu et al. 2004), but we did not attempt to replicate such patterns). The first two types of stimulation – of single pyramidal cells and noisy inputs to LOT – were most informative, and will be used for illustrations here. For each form of stimulation, we tried a number of values for the parameter gAMPA_L2pyr/L2pyr, which scales the strength of (AMPA receptor-mediated) recurrent synaptic excitation between L2pyr (see Methods); this was done while keeping the synaptic connection topology fixed. We varied this single parameter for the sake of conceptual simplicity, realizing that in actual biology many different parameters will be variable, under the influences of past activity, of modulators, and of other factors.
We first undertook an overview of collective behaviors that the model could produce, on a time scale of seconds – the temporal features and qualitative cell compositions. Figure 3 shows the types of behavior that the network model generates, for selected different values of gAMPA_L2pyr/L2pyr. In this figure, based on 5 s simulations, we plot rasters of half of the L2pyr population (cells 1–500), of the deep LTS interneurons, of somatic voltages of selected L2pyr, and of the “field” (inverted average L2pyr somatic voltage). Using a pseudo-random number generator (durand.f), each LOT axon fired a spike at (pseudo)random uncorrelated intervals of mean value 800 ms (1.25 Hz) – a low value for 1 axon but leading to considerable total afferent input in the 500 LOT axons as a whole. The biological information to which this signal might correspond is not specified in our model. As will be seen in subsequent figures, that type of random afferent activity generated action potentials in superficial interneurons, as well as inducing EPSCs in the apical dendrites of all principal cell types – the latter sometimes leading to somatic firing and thus contributing to recurrent stimulation of all interneuron types and of pyramidal cells (there is little recurrent excitation of semilunar cells). Because the “LTS” interneurons (deep low-threshold spiking interneurons, see Methods) do not receive bias currents (hence are not hyperpolarized) and because each such interneuron receives synaptic excitation from 60 L2pyr, the LTS population acts a “sensor” of transiently synchronized L2pyr firing – a cell assembly detector, so to speak.
What is clear in Figure 3 is the emergence of synchronized firing in subsets of L2pyr (marked with red asterisks, and called here cell assemblies), at seemingly random times, and during which L2 fire single action potentials or spike doublets in the middle and right panels (insets) (However, bursts riding on a depolarizing envelope can occur when recurrent excitation is sufficiently large – c.f. Figure 7). In vivo, intracellular recordings of piriform cortex pyramidal neurons exhibit singlets and doublet action potentials in response to odor stimulation (Wilson 1998 – a study with urethane anesthesia), as in Figure 3. Corresponding findings after electrical stimulation of LOT have been described in vitro (Sheridan et al. 2014).
Of note is something that the model does not produce: the development of persistent reverberating activity, as can happen in piriform cortex in vivo and which appears perhaps to be related to working memory (Bolding et al. 2020; Miyashita 1988; Zhang et al. 2019). It is possible that the model’s omission of certain deep principal neurons, such as endopiriform cells, may be the reason for this (Hoffman and Haberly 1993), as could the omission of interactions with posterior piriform cortex, with olfactory bulb (Zhou et al. 2016), and with interconnected cortical regions such as orbitofrontal cortex.
Another interesting feature of Figure 3 is that, in all panels, many of the L2pyr do not fire at all, a phenomenon that is observed in vivo in response to particular odor stimulation (Litaudon et al. 2003). For example, in the left panel of Figure 3, only 120 out of 500 examined L2pyr fire after the initial transient. The relatively infrequent generation of cell assemblies may reflect the “preference” of L2pyr firing to occur in response to synchronized afferent excitation (Luna and Schoppa 2008) which, with our random afferent stimulation paradigm, only occurs on occasion, and at times that are difficult to predict. With pseudo-random stimulation, synchronized excitation of principal neurons could at times also be accompanied by synchronized excitation of superficial interneurons (Pouille et al. 2009), complicating the question of just when principal neurons will discharge.
One test of the model is its ability to generate potentials that resemble sharp waves, as have been recorded in the in vivo olfactory cortex (see for example Figure 1A, B of Manabe et al. 2011). While we are not aware of intracellular recordings during in vivo olfactory cortex sharp waves, it is possible to observe a similar phenomenon in the olfactory cortex slice in the presence of carbachol (5 µM) (Figure 4). The local field potentials (LFPs) in vitro resemble those recorded in vivo. During the in vitro sharp waves, L2pyr generate action potentials and large synaptic potentials – these have a quite similar appearance to model signals, when recurrent L2pyr/L2pyr synaptic connections are powerful enough (Figure 4). As postulated to be the case for sharp waves in vivo in hippocampal CA3 (Buzsáki 1986), the simulated sharp waves depend on recurrent synaptic excitation (Note, however, that in pathological neocortex, sharp waves can also depend on GABAergic mechanisms – see Figure 2 of Roopun et al. (2010)).
The detailed mechanisms by which carbachol induces olfactory cortex sharp waves may be somewhat different than in the model. For example, Hasselmo and Bower (1992) report that carbachol reduces “intrinsic”, that is recurrent, synaptic excitation. In our in vitro experiments, we found that carbachol reduced the amplitude of spontaneous EPSPs but increased their frequency (details in Methods). One interpretation is that carbachol effectively increases synaptic coupling between principal neurons indirectly, by altering cellular intrinsic properties, thereby increasing membrane excitability: for example, in olfactory cortex it decreases firing frequency adaptation (Barkai and Hasselmo 1994). We examine this possibility below (see Figure 12) (Another puzzling fact is that carbachol induces gamma oscillations in the dorsal CA3 region of hippocampal slices, rather than sharp waves (Fisahn et al. 1998). The explanation for this difference is not clear; perhaps electrical coupling in the hippocampus is responsible (Traub et al. 2000, 2003b)).
In order to better understand synchronized principal cell firing in the piriform network model (and for comparison with the other major three-layered cortex, hippocampus), we shall consider two contrasting general scenarios: (i) An idealized situation, wherein feedforward inhibition (LOT→superficial interneuron→principal neuron) of principal cell neurons is blocked, with pyramidal neurons then able to generate bursts of action potentials, and (ii) feedforward inhibition is present (as would presumably be the case in vivo during sensory stimulation). Synchronization in these two situations is different. Case (i) has some correspondence with the CA3 hippocampal region in vitro, especially in a minislice, where one would expect little feedforward inhibition impinging on distal pyramidal cell apical dendrites – provided that we additionally postulate that CA3 pyramidal (CA3pyr) interconnections are powerful enough to allow burst propagation from CA3pyr to CA3pyr; the latter has been shown by Miles and Wong (1987) to be the case during pharmacological disinhibition. A consequence of these conditions is that “percolation” can occur from bursting in a single CA3pyr to eventually recruit the entire CA3pyr population, as has been analyzed previously (Miles and Wong 1983, 1986, 1987; Traub and Miles 1991; Traub and Wong 1982). Here we first examine case (i) with the olfactory cortex model, and then contrast it subsequently with the more realistic (for olfactory cortex) case (ii).
As background, note that burst generation has been described in piriform cortex pyramidal neurons (Doherty et al. 2000; Forti et al. 1997; Hoffman and Haberly 1989 (in low Mg2+); Postlethwaite et al. 1998; Suzuki and Bekkers 2006, 2011; Whalley et al. 2005; Wiegand et al. 2011), although many – but not all – of these cases involve epileptic conditions. Additionally, Ca2+ currents do exist in these cells and probably contribute to bursting (Magistretti et al. 1999; Mukherjee and Yuan 2016; Suzuki and Bekkers 2006). The presence of such Ca2+ currents is important in interpreting the bursting behavior of pyramidal neurons in the next simulations.
In the next two figures, we undertake a detailed comparison of the properties of our piriform model with previous models of network bursting in the in vitro hippocampal CA3 region (Traub and Miles 1991). Figure 5 illustrates simulated network behaviors for three values of gAMPA L2pyr/L2pyr under conditions when superficial synaptic inhibition to L2pyr is blocked. Under such conditions, a sufficiently large somatic current pulse, delivered into a single L2pyr evokes a burst of firing (red traces in the upper panels, marked with a red asterisk in the raster plots). When gAMPA L2pyr/L2pyr = 8 nS (Figure 5A), only a few L2pyr are recruited (and only a few basket cells), generating a small cell assembly. Repeating with a slightly larger gAMPA,L2pyr/L2pyr of 9 nS (Figure 5B) allows percolation from the single stimulated cell to the entire population, leading to a synchronized discharge – a result strikingly similar to data from the CA3 region in vitro (c.f. Figure 9E of Miles and Wong (1987) and Traub and Wong (1982)); however, in the present case (contrasting with earlier hippocampal experiments), GABAA-receptor mediated inhibition is not completely blocked, rather only superficial inhibition. Finally (Figure 5C), another further increase of gAMPA L2pyr/L2pyr to 11 nS allows the stimulated single-cell burst to lead to a synchronized discharge at shortened latency. This simulation makes a prediction which could lend itself to experimental test, although with an important proviso (see Discussion): the unitary excitatory synaptic conductances in this model may be quite a bit larger than in vivo, or at least larger than in vivo mean excitatory synaptic conductances. Thus, in the olfactory cortex – unlike the CA3 region in vitro (Miles and Wong 1987) – it may or may not be possible for bursting to spread from a single pyramidal cell to other pyramidal cells; in piriform cortex, stimulation of some critical number of pyramidal cells may therefore be necessary to cause a synchronized population burst.
Continuing our comparison of the olfactory cortex model with the hippocampal CA3 region in vitro, we note that recurrent inhibition within CA3, mediated in part by fast-spiking interneurons, is both rapid and powerful (Miles and Wong 1984, 1987): disynaptic inhibition from one CA3pyr to another is initiated faster than monosynaptic excitation (mean latency of disynaptic IPSP 3.1 ms (Miles and Wong 1984); unitary monosynaptic CA3pyr/CA3pyr EPSPs peak at 7–12 ms (Miles and Wong 1986)). This striking circuit property depends in part on the ability of a single CA3 pyramidal cell action potential to initiate an action potential in a monosynaptically connected interneuron (Gulyás et al. 1993; Miles 1990). Whether this particular circuit motif exists in piriform cortex is, to the best of our knowledge, not known.
With the particular parameters that we have chosen, the behavior of our network model is different than in vitro CA3 (Figure 6). Thus, blocking both superficial (as in Figure 5) and deep synaptic GABAA inhibition to L2pyr, and exciting a single L2pyr, one observes an intense synchronized L2pyr discharge (Figure 6A, gAMPA L2pyr/L2pyr = 8.5 nS). This is not surprising. However, restoring deep inhibition and exciting the same single L2pyr (Figure 6B) does not prevent the occurrence of a synchronized cell assembly; instead, a cell assembly occurs, although one much less intense than before. When the strength of the unitary L2pyr → basket (AMPA receptor mediated) conductance was doubled (Figure 6C), allowing the earlier firing of some basket cells (visible in the raster, red arrow), the size of the cell assembly is noticeably reduced; however, a cell assembly still occurs. Experimental measurement of the unitary L2pyr/L3pyr → basket EPSP will help to constrain this aspect of the model. It is important to emphasize that spread of bursting in the simulations of Figures 5 and 6 represents synaptic interactions predominantly between pyramidal neurons, involving the properties of the recurrent connections and intrinsic membrane properties; in contrast, in vivo, afferent inputs will be present as well, possibly combining in pyramidal neurons with recurrent and intrinsic sources of excitation.
We now turn to a more physiological – but still idealized – situation in which superficial inhibition is present, and LOT fibers generate, over a defined time interval, random-time Poisson-distributed action potentials. A number of simulations were run; these were identical except for the variation of gAMPA L2pyr/L2pyr (Figure 7). Noting that total L2pyr activity increased as gAMPA L2pyr/L2pyr was increased, while holding all other parameters constant, we sought to quantify the activity increase. This was accomplished by defining an “order parameter” (a statistical physics term) equal to the number of L2pyr axonal action potentials (for L2pyr 1 … 500) within a fixed time interval (time = 750–800 ms, an interval in which peak synchronized activity occurred). The idea was to see if ideas taken from statistical physics could be used to understand the evolution of collective behaviors in the model, as a structural parameter was varied.
As shown in Figure 7, depending on the excitatory synaptic coupling between L2pyr cells, the system exhibits three distinct behaviors (phases) that can be characterized by this order parameter. For small coupling strengths, the firing pattern seems random without occurrence of an assembly (c.f. Figure 3, left panel, for an example, with gAMPA L2pyr/L2pyr = 6 nS); therefore, the order parameter in this random-firing phase is near zero. As the coupling increases past a critical value (∼8 nS), the order parameter becomes finite, indicating the onset of collective cell assemblies. At a still higher critical value (∼11.5 nS), events resembling sharp waves appear and some L2pyr cells generate burst firing patterns. The first transition from the random-firing phase to the assembly phase resembles a second order phase transition as studied in statistical physics. For example, for a system of noisy coupled oscillators, it was shown that the system changes from an asynchronous phase to a synchronous phase through a second order phase transition where the synchronization order parameter changes continuously from zero to finite values as the coupling strength increases across a critical value (Kuramoto 1975, 1984). At the second transition from cell assemblies (Figure 7 inset 1) to sharp wave-like events (Figure 7 inset 2), the firing behavior of L2pyr transforms from single spikes and/or doublets to bursts that ride on a depolarizing envelope, in some cells; this bursting resembles epileptiform behavior (A similar type of behavior for hippocampal CA3 is discussed in Traub and Miles (1991), although in that case, the model network was disinhibited, unlike the present case). This transition, to sharp waves, is characterized by a large abrupt increase in the order parameter, which measures the total axonal activity within a specified time interval. This abrupt change suggests that the cell-assembly to sharp-wave transition may be analogous to a first order phase transition (such as melting of ice or boiling of water), where the order parameter changes discontinuously at the critical point. Due to the finite-size model and limited number of simulations, we do not wish to make rigorous claims regarding the nature of these transitions here. However, analogies to phase transitions studied in statistical physics (Cross and Hohenberg 1993) may provide a fruitful direction for future investigation.
The piriform cortex contains a number of different types of interneurons, some predominantly feedforward, others predominantly feedback. In the next two figures, we use the model to analyze the differential responses of interneurons. During random LOT stimulation of the sort used to construct Figure 7, superficial and deep model interneurons behave quite differently, especially in the regime before epileptiform activity develops. Thus (Figure 8, gAMPA L2pyr/L2pyr = 8 nS), superficial interneurons are quite active, while basket cells are hardly active at all, even as a recognizable cell assembly appears; both are consequences of connectivity and other parameter choices, including the smaller bias currents to basket cells as compared to LTS cells (Methods). All principal neurons exhibit prominent ongoing synaptic potentials (three L2pyr somata are shown in Figure 8). With stronger L2pyr coupling (gAMPA L2pyr/L2pyr = 11 nS, Figure 9) and the emergence of a more intense L2pyr cell assembly (that is, with L2pyr firing), basket cells are then prominently recruited. Note in Figure 9 how the cell assembly appears to emerge (in the L2pyr raster), growing out of action potentials and spike doublets in a collection of scattered L2pyr (horizontal blue arrows). This type of buildup, from multiple neurons, stands in contrast to the idealized situation illustrated in Figure 5, where growth occurs from a single neuron (red voltage trace, red asterisk in L2pyr raster).
The generation of firing activity of a cell assembly occurs over time; the details of firing activity may be different for cells firing early compared with cells firing late. The next figure shows an example of this. Figure 10 illustrates details of the buildup for a case similar to that illustrated in Figure 9 (gAMPA L2pyr/L2pyr = 11.5 nS in Figure 10 vs. 11.0 nS in Figure 9). In Figure 10, a robust cell assembly (resembling a sharp wave) emerges from a small population of L2pyr. The system now appears to be acting as an associative memory, whereby activation of one set of principal neurons reliably recruits another set via recurrent excitatory connections (Barkai et al. 1994; Bolding et al. 2020; Guzman et al. 2016; Hagiwara et al. 2012; Hasselmo and Barkai 1995; Katori et al. 2018; Palm 1980; Pashkovski et al. 2020; Yang et al. 2004) – if a rather exuberant one. An issue for the future is to determine how and when the recruited neurons themselves start to recruit additional neurons – presumably, a disruption of associative memory function.
Because LOT afferents excite both principal cell and superficial interneuron dendrites, the effects of LOT activity are not easy to predict, especially with random LOT spikes as we typically use. Figure 11 demonstrates that the relation between LOT intensity and cell assembly formation (other things being equal) is not straightforward: at least over a certain parameter range, cell assemblies can become more intense (that is, have more pyramidal cells recruited) with a reduction in LOT intensity (Figure 11A vs. Figure 11B). This behavior could arise because of varying effects on direct excitation of pyramidal dendrites versus excitation of superficial interneurons and feedforward inhibition. Of course, with sufficiently sparse LOT activity (Figure 11C), assembly formation attenuates. Further examination of this issue will require a proper accounting of temporal correlations in LOT activities (Apicella et al. 2010; Avorio et al. 2019; Haddad et al. 2013), in contrast to the uncorrelated conditions simulated here.
We have raised the issue above (in our discussion of sharp wave-like events and the work of Hasselmo and colleagues) that effective coupling between principal neurons can be influenced by intrinsic membrane conductances (even with constant synaptic conductances), and this could be especially important in interpreting the effects of cholinergic agonists on piriform circuits. For this reason, we examined the role of the slow Ca2+-mediated AHP (afterhyperpolarization) current in determining the intensity of cell assemblies (Figure 12), modifying the AHP properties in L2pyr only and leaving all other parameters (specifically including gAMPA L2pyr/L2pyr) fixed. As expected, and alluded to above, a reduction in AHP conductance increased the effective excitatory synaptic coupling between L2pyr and led to larger cell assemblies, while an increase in AHP conductance had the opposite effect. Both increases and decreases in AHP conductances could lend themselves to experimental tests (Barkai and Hasselmo 1994; Suppes et al. 1985).
Understanding how the piriform cortex responds to odors requires consideration of collective neuronal behavior, given that responses are highly distributed across the structure (Illig and Haberly 2003; Rennaker et al. 2007; Roland et al. 2017). Our main simulation prediction is that piriform cortex generates three general sorts of collective behavior, regulated in part by the strength of synaptic coupling between pyramidal cells: noisy appearing background, synchronized cell assemblies with single spikes and doublets in some pyramidal neurons, and sharp wave-like behavior with synchronized bursting in many of the pyramidal neurons. While it is difficult experimentally to account precisely for how downstream areas (orbitofrontal cortex, amygdala, lateral entorhinal cortex, etc.) would respond to this or that collective piriform activity, the details of such activity must matter: spatially sparse distributed ensembles of olfactory cortex neurons (as determined by Fos tagging) encode olfactory fear conditioning, in a way that is functionally relevant, as determined by the effects of silencing such ensembles (Meissner–Bernard et al. 2019). Evidently it is important to better characterize how ensemble activity is produced, the goal of the present study, although here we concentrate on temporal aspects rather than spatial ones.
In the present study, we have chosen to focus on the ability of a model network, with certain parameter choices, to generate cell assemblies. The generation of assemblies is a robust feature of the model, provided recurrent excitatory synapses are powerful enough, and chemical synaptic and intrinsic (gK-mediated) inhibition are not too strong. It is critical to note, however, that – as Figure 7 shows (points to the left of point 1) – the development of synchrony occurs gradually with increasing recurrent excitation, at least until a critical level of recurrent excitation is reached. Therefore, the decision is arbitrary on when to call a certain degree of synchrony a “cell assembly”. We suspect that point 1 would qualify. To the right of the inflection in the graph of Figure 7, say point 2, network behavior is clearly different. Whether to call this type of behavior a sharp wave, or possibly an epileptiform burst, should be determined (we believe) by in vivo experiment: specifically, by intracellular recordings of pyramidal cell firing during putatively normal sharp waves, as illustrated by Manabe et al. (2011), and during interictal bursts in animals that also demonstrate behavioral seizures. We note, however, that a number of authors, working in vivo, have suggested that sharp wave-like potentials in the piriform cortex have properties similar to physiological hippocampal sharp waves (Barnes and Wilson 2014; Narikiyo et al. 2014): in each case, the potentials propagate from one structure to another, and may play a role in memory consolidation.
Is it then the case that cell assemblies, however one chooses to define them, are the critical (that is, biologically meaningful) feature of any given simulation? While this is ultimately an experimental question, simulations can help. One important value of the simulations is that by identifying cellular mechanisms that can generate cell assemblies, one can next devise experiments that perturb these mechanisms, and determine what, if any, are the behavioral consequences. A similar approach has been used in the study of hippocampal sharp waves and their replay properties (Girardeau et al. 2009), and of hippocampal gamma oscillations (Fuchs et al. 2007). If, as one expects, future experiments support a behavioral significance of piriform cell assemblies, then one can guess that the temporally focused firings of defined subsets of neurons are what carry the important messages out of piriform cortex, while the relatively (temporally) uncorrelated background is ignored by downstream regions. In the case of hippocampal sharp waves, there is considerable fine structure, and apparently functional significance, not only in the identity of cells firing during the sharp wave but in the details of their relative firing times within the sharp wave (Diba and Buzsaki 2007). While the mechanisms that determine these firing times are not completely settled (Traub et al. 2020), the hippocampal data are a useful point for comparison in future modeling and experimental studies of piriform cortex.
A similar logic of relating cellular mechanisms to network activity and sometimes to in vivo observable behavior has been pursued with respect to one type of gamma oscillation. Discussion of this logic may be helpful in conceiving experiments for piriform cortex. There are conceptual reasons to suspect that oscillations may be important functionally in that they produce temporally coincident action potentials in different neurons (say A and B), allowing neuron C – which receives input from A and B – to detect the temporal coincidence (Konig et al. 1996). A notion like this could – in principle – apply to piriform cortex, even in the absence of oscillations. Note as well that sparse background activity, such as our model exhibits with weak coupling between pyramidal cells, would not be expected to produce temporal coincidences – although it should be recalled that in other experimental paradigms, pyramidal cell coincidences can occur with minimal or no recurrent excitation between principal neurons (Traub et al. 2000; Whittington et al. 1995). For synchronized gamma oscillations induced in the CA1 region in vitro by two-site tetanic stimulation, it was predicted that large rapid-onset rapidly decaying EPSCs in fast-spiking interneurons were critical for oscillation synchrony (Traub et al. 1996; Whittington et al. 1997; reviewed in Traub et al. 1999). As the kinetic properties of interneuron EPSCs depend on AMPA receptor subunit composition, the data motivated a genetic manipulation of AMPA receptor subunit composition in fast-spiking interneurons (Fuchs et al. 2001), which was shown to lead to the predicted consequences for oscillation synchrony. In analogous fashion, it might be possible, using in vivo molecular tools, to manipulate AMPA receptors in recurrent L2pyr connections without affecting the LOT inputs. Such a manipulation could have both electrophysiological and behavioral consequences. One study in this direction has been performed by Shimshek et al. (2005), who noted that manipulations of GluR-B (GluR2) – an AMPA receptor subunit found predominantly in principal neurons (Racca et al. 1996) – affected olfactory discrimination and learning and memory.
A complimentary in vitro approach could be as follows. To test the main thesis of this study (e.g. as in Figure 7) one would like to enhance the strength of recurrent excitatory pyramidal/pyramidal synapses without simultaneously altering properties of the afferent synapses. This is likely to be technically hard. However, using the approach illustrated in Figure 12, one could possibly enhance, in graded fashion, the effective excitatory synaptic coupling by slowly blocking pyramidal cell gK(AHP), while at intervals delivering constant afferent stimulation and recording intracellular and field potentials. Our model predicts a first gradual increase in field potential amplitude, followed by a strong upward inflection.
Other authors have constructed circuity models of olfactory cortex, concentrating on different, if overlapping, aspects of collective neuronal behavior. Of particular interest, Stern et al. (2018) were concerned with spatiotemporal patterns of piriform responses to realistic afferent input, especially issues of the timing of inputs and corresponding responses, and the effects of concentration changes of odors. All model cells were integrate-and-fire (pyramidal cells, superficial and deep interneurons), making it practical to build a large model, for example with 10,000 pyramidal cells and 1000 recurrent inputs per cell (so that each recurrent input could be weaker than in our present model). Patterns of activity were analyzed in terms of firing rates, rather than the coordinated firing associated with a cell assembly. The type of model employed by Stern et al. lends itself to the study of complex spatial patterns of input and spread, but does not lend itself to the role of dendritic and somatic intrinsic properties of the cells (like AHPs, burst generation), or of more complex synaptic properties (perisomatic vs. dendritic, interactions between synaptic inputs arriving at distinct neuronal compartments). The model of Stern et al. (2018) was also not designed to study the transition from normal to epileptiform activity, or the fine structure of sharp waves.
The comparison of the present study with the approach of Stern et al. (2018) raises a question that is technical, but still important: what is the appropriate level of complexity to incorporate in single-cell models, used to study population phenomena (Kopell 2005; Pinsky and Rinzel 1994)? Probably no definitive answer exists to this question but our opinion is this: clearly, for piriform cortex, certain structural features are likely useful, such as incorporating enough dendritic length and dendritic intrinsic conductances (such as gCa) to allow for bursting and to capture the laminar arrangements of synaptic connectivity. If electrical coupling turns out to be important in piriform cortex, then axons may also need to be included (Traub et al. 2018). On the other hand, it is likely that simpler model neurons could be used, simpler than we have employed, with fewer compartments and fewer active conductances. The choice for the modeler is then between employing modeling tools that exist already, or can be easily adapted (as we have done), or rather to develop new tools de novo.
In earlier modeling studies, Wilson and Bower (1992) used a 4500 cell network (5-compartment pyramidal cells, 1-compartment interneurons) aimed mainly at the analysis of local field potential and EEG oscillations, and emphasizing a putative role of axonal conduction delays. Barkai et al. (1994) and Hasselmo and Barkai (1995) concentrated on cholinergic effects on pyramidal cell intrinsic properties and on synaptic transmission and plasticity, and on the implications thereof for learning and pattern completion.
Limitations imposed by the small size of model
Srinivasan and Stevens (2018) estimated that mouse piriform cortex, anterior together with posterior, contains about 500,000 neurons. The present model has only 2700 neurons, more than 100-fold smaller than the Srinivasan and Stevens estimate, although we are modeling anterior cortex only. It is possible, therefore, that the size of recognizable cell assemblies, relative to total population size, will be larger in the model than in vivo – perhaps much larger. Furthermore, we use 500 LOT axons, but, in contrast, rat LOT has about 30,000–40,000 axons, depending on the anatomical section; and with the considerable branching of LOT axons, the effective connectivity will be still larger than the axon counts suggest (Price and Sprich 1975).
In addition, the small model network size forces us – in order to see any collective behavior – to use unitary synaptic conductances that are perhaps unrealistically large (but see below). Thus, Franks et al. (2011) state that pyramidal cells are “sparsely interconnected by thousands of excitatory synaptic connections.” They write that pyramidal cell axons project across the piriform cortex, but contact <1% of other cells; and that each cell receives input from >2000 other pyramidal cells. Similarly, ul Quraish et al. (2004) report that a single L2pyr contacts 2200–3000 other pyramidal cells. In our model, each L2pyr receives input from 20 other L2pyr (Methods), or 2% of the L2pyr population – not too far off, but the actual number of inputs is about 100-fold less than in vivo. Conversely, the unitary synaptic conductances we use are large. Franks et al. (2011) estimate unitary excitatory postsynaptic currents (EPSCs) in pyramidal cells to be in the range of ∼25–35 pA. Our own slice data, with sharp electrodes, suggests a value of about 20 pA. In the model, with a typical gAMPA L2pyr/L2pyr of, say, 10 nS, then a unitary EPSC would be about 10−8 S × (2/e) × 70 × 10−3 V ∼ 50 × 10−11 A = 500 pA, or ∼25-fold too large (balancing, one notes, the reduced connectivity).
One important – indeed critical – possibility, however, is that some of the recurrent excitatory synaptic conductances are large, even if most are not. There is a precedent for this in the excitatory synaptic connections between large layer 5 tufted pyramidal cells in young rat neocortex: Unitary EPSPs could range from 0.15 to 5.5 mV (Markram et al. 1997). If a similar distribution were to exist between L2pyr – something that would need to be demonstrated experimentally – then our model could be seen as incorporating one piece of a skewed conductance distribution and ignoring the rest.
Scaling issues also arise on the number of afferents in the model. Thus, Davison and Ehlers (2011) report that a piriform pyramidal cell receives about 200 afferent inputs from the olfactory bulb versus only 20 in the model. This type of connectivity will be important if, as Davison and Ehlers write, pyramidal cells are sensitive to patterns of bulbar glomerular activation, rather than activation from a single glomerulus (but see Franks and Isaacson (2006) for a somewhat different view). Presumably, it is particular temporal correlations in afferent input, as well as the recurrent connectivity, that trigger the appearance of cell assemblies. The noisy LOT input in most of our simulations omits temporal correlations between mitral cells (Buonviso et al. 1992) – a factor that will influence the unpredictable timing of cell assemblies, and perhaps their size. It is possible, on the other hand, that variability in the timing of cell assemblies reflects the actual biology of the system, as observed in the significant variability of olfactory response times in human subjects (Stevens et al. 1988).
Scaling issues of this sort often arise whenever a small circuit is used to model a large one. Sometimes, however, a basic physiological principle emerges even with a small model. For example, Traub and Wong (1982) used a model with only 100 neurons to study epileptiform behavior in the disinhibited CA3 region in vitro. Experiments showed that a focal stimulus would reliably recruit the entire neuronal population, with a latency of order ∼100 ms. It was then easy to predict some fundamental properties of the system: that a burst of action potentials in one pyramidal cell should induce bursting in a follower cell, and that each pyramidal cell should synaptically excite more than one other pyramidal cell, on average. In this situation, a chain reaction could occur. Both predictions were experimentally verified (Miles and Wong 1987). The present situation is far more complex, however. Afferent inputs are present, and synaptic inhibition not only exists but takes on a complicated form because of the multiple interneuron types, each with its own connectivity. For these reasons, the results of a relatively small model need to be taken as preliminary hypotheses.
How else can the possible role of cell assemblies in olfactory cortex be further examined? In terms of experiments, two types of study seem important. First is the actual detection of cell assembly-like activities in vivo, using multiunit data (Lopes-dos-Santos et al. 2013). Second is the characterization of unitary synaptic interactions between piriform principal neurons, using dual (or multiple) simultaneous intracellular recordings. In terms of modeling, the approach of Stern et al. (2018), that is with integrate-and-fire neurons, is possible on a large scale, but suffers from the disadvantage of omitting intrinsic membrane currents (gCa, gK(AHP)) that seem important, as well as omitting dendritic synaptic integration, possibly NMDA spikes (Kumar et al. 2018) (which we ourselves have not yet examined) and important modulator effects on all of these properties. An alternative would be to scale up the present model to the extent that is computationally practical.
Implications for epileptiform activity in olfactory cortex
The piriform cortex is especially prone to epileptiform behavior in humans (Vaughan and Jackson 2014) as well as in a number of experimental preparations (de Curtis et al. 1994, 2019; Doherty et al. 2000; Forti et al. 1997; Galvan et al. 1982; Haberly and Sutula 1992; Hamidi et al. 2014; Löscher and Ebert 1996; Panuccio et al. 2012; Pelletier and Carlen 1996; Uva et al. 2017; Whalley et al. 2005; Young et al. 2019). This appears to be especially the case for the circuits deep to layer 3 (not modeled here), including endopiriform cortex (Hoffman and Haberly 1993; Piredda and Gale 1985). Patients with focal seizures originating in temporal lobe may have olfactory abnormalities (Desai et al. 2015) and olfactory stimuli can sometimes initiate seizures (Avorio et al. 2019) (Olfactory auras, however, appear to mostly originate in the amygdala (Acharya et al. 1998; Chen et al. 2003; Lee et al. 2009)). Our data do not shed new light on epilepsy mechanisms per se, but do emphasize that the generation of cell assemblies, with some degree of synchronization, might – with relatively small parameter changes and under some circumstances – switch to the full synchronization characteristic of an epileptiform discharge. Indeed, this is one of the major findings of our study. It seems reasonable to draw an analogy between (normal) physiological sharp waves in the hippocampus (Buzsáki 1986; Csicsvari et al. 2000) and epileptiform discharges there.
In both cases, olfactory cortex and hippocampus, a fundamental question is this: there must be some essential “payoff” in designing circuitry with enormously rich recurrent excitation, at the cost of possibly predisposing toward epileptic seizures. Proper analysis of this problem is not straightforward. Thus, the generation of highly synchronous cell assemblies, or sharp waves, may not in itself be harmful; indeed, such activities could well be essential for normal brain function. In that case, the transition to frankly epileptiform activity requires additional perturbations besides strong synaptic coupling between pyramidal neurons. The question still remains, however, why it is that extreme richness of recurrent excitation occurs in some brain areas but not others?
Comparison with other structures and species
We have contrasted the properties of a model of mammalian anterior piriform cortex with mammalian hippocampus (as studied in vitro). The matter of interest is how recurrent excitatory synapses shape cell population behavior. This issue may have wide ramifications, given observations in other systems and other species. For example, Houweling and Brecht (2008) used juxtacellular stimulation of single mammalian somatosensory cortical neurons; they observed that such stimulation could have behavioral consequences, something only possible through spread of activity – although not necessarily within the local circuit (Acharya et al. 1998; Ainsworth et al. 2012). Hemberger et al. (2019) observed, in turtle cortex (three-layered, as is mammalian olfactory cortex) that two action potentials, induced in single neurons, could induce reliable sequences of action potentials, in tens of neurons, over a time interval up to 200 ms. Thus, consistent with ideas advanced by many others, the proposed cell assemblies in piriform cortex may occur in a wider context in vertebrate brains.
A recent paper (Schoonover et al. 2021) shows that the structure of piriform cortex responses (in vivo) to odors evolves on a time scale of days to weeks. This suggests that the role of recurrent excitatory connections, in contributing to cell assembly formation, is plastic according to mechanisms yet to be determined – thus adding an additional level of complexity to the analysis of cell assemblies.
In summary, our detailed computational model of a piece of anterior piriform cortex predicts the occurrence of synchronized pyramidal cell activity, an activity that may be designated cell assemblies. The assemblies may occur in the presence of random patterns of afferent input, and the number of cells participating – as well the firing patterns of individual neurons – depends strongly on the strength of recurrent excitatory synaptic coupling, as well as on intrinsic cell properties such as AHP conductances. The presence of an assembly may be “detected” by a population of interneurons, if those interneurons are excitable enough. With random afferent input, the times at which assemblies occur are somewhat unpredictable. Assemblies of the sort predicted here would send powerful messages to downstream cortical/amygdalar areas.
Funding source: IBM Exploratory Research Councils
We thank Ben Strowbridge, Carl Edward Schoonover and Andrew Fink for helpful discussions and sharing unpublished data; Sam Mckenzie for helpful discussions; Robert Walkup for critical assistance with programming issues.
Author contributions: All authors conceived the study and wrote the paper. RDT performed simulations. YT performed theoretical analysis. MAW acquired and analyzed experimental data.
Research funding: IBM Exploratory Research Councils.
Conflict of interest statement: The authors declare that no conflict of interest exists.
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