High-fidelity audio recording and playback technologies use a minimum standard sampling rate of 44.1 kHz. How auditory neuron ensembles cope with this sampling rate was not clear until recently, when a computational neuroscience model illustrated periodic pitch perception on a microsecond scale (Klefenz and Harczos, 2020). In the model, equilibrium tipping points are evoked by an excess of excitatory vesicles relative to the currently available reservoir of inhibitory vesicles at the soma of inferior colliculus (IC) neurons and recorded with extreme temporal precision. Oscillations of octopus neurons are perceived by IC neurons as differentiable pitch sensations. Sounds are transformed into spike-based event representations by a bio-plausible, neuro-physiologically parameterized auditory model (Harczos, 2015; James et al., 2017; Cramer et al., 2020; Gutkin, 2020; Baby et al., 2021; Gutierrez-Galan et al., 2021; Saddler et al., 2021). First, the auditory model computes spike train patterns for auditory nerve fibers (ANFs). The auditory nerve divides into several sub-nuclei of the cochlear nucleus. In the dorsal cochlear nucleus, octopus cells receive ANF spike trains in their tonotopically arranged local receptive fields (Kane, 1973; Spencer et al., 2018). The model calculates the periodicity of pitch from the rhythmic oscillations of the octopus cells. The inter-spike intervals (ISI) of octopus cells are measured by batteries of interval-tuned neurons (ITNs) by encoding the interval durations as first spike latencies (FSLs) (Aubie et al., 2009, 2012). Aubie's model is formulated in NEURON with excitatory NMDAR/AMPAR GABAergic inhibition¹ (Kopp-Scheinpflug et al., 2018) and has been adapted and optimized to work reliably in the microsecond range (Klefenz and Harczos, 2020). For better understanding, some parts of the two articles in the Periodicity Pitch Perception series (Harczos and Klefenz, 2018; Klefenz and Harczos, 2020) are recapitulated and some of the figures are reprinted.

Synaptic plasticity depends on its dendritic location, the detailed timing protocol of pre- and postsynaptic events, and the temporal states of postsynaptic hyper/depolarizations (Bach and Kandler, 2020). STDP signaling cascades enlarge dendritic spines through polysome association (Wierda et al., 2020). Initially, the synapse is preactivated by the release of neurotransmitters into the synaptic cleft when triggered by an ANF spike and neurotransmitters open (NMDA) channels for Ca₂₊ influx. Subsequently, STDP requires postactivation, which occurs through the generation of a somatic sodium (Na₊) spike that propagates back into the dendrites (Winters and Golding, 2018). The briefly previously presensitized synapse is potentiated by the passing backpropagating spike, as these concurrent pre/post stimuli trigger the influx of calcium into the spike head of the synapse (Franzen et al., 2020; Kladisios et al., 2020). If it is an inhibitory synapse, it will be suppressed. If a group of synapses is activated by presynaptic events but does not elicit a somatic spike, its weighting is lowered because the postcondition for potentiation is absent. The membrane voltage at the synapse can be modeled as an algebraic equation based on the summation of an excitatory postsynaptic potential (EPSP) and a backward propagating action potential (bAP) (Jahr and Stevens, 1993; Griffith et al., 2016). The stimuli are membrane depolarization due to an EPSP and a bAP about 2 ms apart. When the EPSP arrives 2 ms before the bAP, the maximum possible membrane depolarization is elicited (Hu and Bean, 2018). In general, a single general plasticity rule is sufficient to reproduce different results of plasticity experiments at different dendritic sites, allowing unification of classical STDP- and Ca₂₊-based rules. The plasticity rule can be easily combined with detailed neuron models to study both STDP and plasticity mediated by dendritic Ca₂₊ and Na₊ spikes, NMDA spikes, and synaptic cluster activation (Palmer et al., 2014; Foncelle et al., 2018; Augusto and Gambino, 2019). When a pre-post spike pair is insufficient to trigger potentiation, spike triplets are. The solution approach combines dendritic back-propagation with triplet spike timing dependent plasticity signaling. Potentiation is, therefore, possible when isolated spike triplets are present (pre-post-post or post-pre-post). A pre-post-post protocol triggers much more post-LTP than a post-pre-post protocol. Synaptic potentiation is triggered by spike triplets consisting of one presynaptic and two postsynaptic spikes (Pfister and Gerstner, 2006). Synaptic plasticity is discussed with special emphasis on the role of NMDAR and AMPAR signaling cascades (Rajani et al., 2020). The relative position of postsynaptic AMPAR domains with respect to presynaptic release sites and the molecular basis of such co-organization have been investigated in several studies (Goncalves et al., 2020). Bell et al. (2019) model spiny heads along dendrites by boundary conditions at the plasma membrane (PM) and spiny apparatus (SpApp) in a spatial multicompartment reaction-diffusion model of calcium dynamics in three dimensions with different flux sources, including N-methyl-D-aspartate receptors (NMDARs), voltage-sensitive calcium channels (VSCCs), and various ion pumps at the plasma membrane (PM) (Ingólfsson et al., 2017; Cheng and Smith, 2019; Ohadi et al., 2019). AMPA-type glutamate receptors (AMPARs) mediate fast excitatory synaptic transmission (Choquet, 2018). AMPARs are concentrated within the postsynaptic density (PSD) in small nanoclusters of approximately 80 nm in size, containing an average of 20 receptors (Masugi-Tokita et al., 2007; Fukata et al., 2013; MacGillavry et al., 2013; Nair et al., 2013). Because of AMPAR's low glutamate affinity glutamate must be released precisely in front of AMPAR nanoclusters and, therefore, the relative positioning of pre-synaptic AMPAR‘s release sites with respect to AMPAR nanoclusters is the critical factor for synaptic transmission (Choquet and Hosy, 2020). Active glutamate release sites are co-localized with the presynaptic active zone protein RIM and aligned with AMPAR nanoclusters (Beique et al., 2006). To model the effects of delocalization of AMPAR nanoclusters from presynaptic glutamate release sites, Haas et al. (2018) performed Monte Carlo-based simulations using the MCell/CellBlender simulation environment (http://mcell.org) with MCell version 3.3, CellBlender version 1.1, and Blender version 2.77a (http://blender.org). Kinney et al. (2013) and Bartol et al. (2015b) obtained synaptic shape and peri-synaptic environment from 3D electron microscopy images (Kinney et al., 2013; Bartol et al., 2015a,b). Jonas et al. (1993) determined the chemical kinetic properties of AMPAR using an established model (Jonas et al., 1993), and Nair et al. fitted the kinetic parameters to both the recorded mEPSCs and the AMPAR organization map obtained from d-STORM data (Nair et al., 2013). In their simulations, Savtchenko et al. set the number of glutamate molecules released to 1,500, 2,000, 3,000, or 4,500 to be within the range of the estimated amount per presynaptic vesicle (Savtchenko et al., 2013). The simulations calculated the number of open AMPARs when vesicles containing the different amounts of glutamate were released upstream of a single AMPAR cluster or up to 200 nm away from the cluster center, varied with a step size of 50 nm. Nair et al. (2013) adjusted the AMPAR rate constants in their model using simplex optimization with minimal least squares to best fit the shape of the AMPAR current. Jonas et al. (1993) set the initial parameter values with the release of glutamate directly across the cluster, using n_Glu = 3,000, n_AMPAR = 25 in the cluster. The AMPAR activation time courses of 100 simulation trials were averaged at each release site. Computer modeling predicts that a lateral shift of approximately 100 nm between AMPAR nanoclusters and glutamate release sites results in a significant reduction in AMPAR-mediated currents (best fit parameter values in Kim et al. (2018). In the method part the functional octopus' model will be unrolled. Clues and pointers for hardware realizations of the octopus' cell are stated in the discussion section.

The function of the octopus' cell is to permanently observe spatiotemporal trajectories in its local receptive field and to selectively respond to a trajectory with a specific hyperbolic shape. Constellations of distributed, cascaded synaptic input activations predetermined by the hyperbolic shape of a trajectory can lead to coincidence detection at the soma by triggering a spike for that event. The coincidence detection function of the octopus' cell is based on the morphology of its dendritic tree, the distributions of synaptic inputs along the dendritic trees, the event-based cascades of synaptic input activations and the local signal propagation velocities in the dendritic branches (Remme et al., 2018). ANFs innervate octopus' cells through synaptic connections along their dendritic trees. At the synaptic sites, activation triggers dendritic spikes that propagate to the soma. In this way, even the most distant synapses influence the electrical potential at the soma. In the simplest case, a single distant synaptic input is attenuated and low-pass filtered before it reaches the soma. A single EPSP remains below threshold, and most of the collective synaptic potentials are too weak on their own to trigger a somatic action potential, but some constellations have converging EPSPs that sum at the soma and trigger an AP (Kladisios et al., 2020). The dendritic calculation evaluates the threshold crossing condition as a function of the actual sum of EPSPs arriving simultaneously at the soma. The potential flow calculation can be performed for each local dendrite segment by assigning a cable conductance value. These conductance values determine the local signal propagation velocities. The conductances allow the computation of coincidences of temporally consecutive synaptic inputs within a branch, between different branches, or throughout the dendritic tree (Li et al., 2019). With precisely activated inputs cascaded in time, the potential currents in the dendritic branches converge according to Kirchhoff's current law and swell until they overflow at the soma. An Na₊ or an NMDA (N-methyl-D-aspartate) or a Ca₂₊ ion channel could be one of the transmissive cable lines (Spruston et al., 1995a).

ANF spike trains trigger dendritic spikes at synaptic sites distributed along the dendrite tree. The collaboration of synaptic inputs from multiple dispersed sites is required for somatic spiking (Urbanczik and Senn, 2014). Some spatio-temporal coherent constellations of dendritic spikes arrive synchronously at the soma and reliably generate temporally precise APs and thus event-based timestamps. Complementarily, APs fired at the axon hillock bounce back in the opposite direction, back-propagate along the dendrites and cause postsynaptic depolarizations at the synapses they pass (Hoffman et al., 1997; Magee and Johnston, 1997; Hebb, 2005; Dan and Poo, 2006; Feldman, 2012). The back-propagating action potential (bAP), therefore, satisfies the post-signaling condition of STDP (Levy and Steward, 1983). The learning signals of bAPs resemble backpropagation through time (BPTT) with surrogate gradients and target-prop algorithms (Werbos, 1990; Sacramento et al., 2018; Neftci et al., 2019; Lillicrap et al., 2020). Learning in dendritic regions distant from the soma is problematic because bAP does not reach the most distal dendrites or reaches them only in an attenuated manner. Therefore, a bAP is unlikely to provide the necessary depolarization to contribute to the induction of LTP at the most distal synapses (Krueppel et al., 2011). Plasticity can also be triggered by depolarizations originating from sources other than bAPs in the postsynaptic neuron, e.g., dendritic Ca₂₊ spikes (Golding et al., 2002; Kampa et al., 2006; Letzkus et al., 2006), N-methyl-D-aspartate (NMDA) spikes (Gordon et al., 2006; Brandalise et al., 2016), or excitatory postsynaptic potentials (EPSPs) alone for LTP induction at the most distal synapses (Golding et al., 2002; Lin et al., 2008; Sjöström et al., 2008; Weber et al., 2016; Kim et al., 2018). Although many biophysical details of excitatory synaptic long-term plasticity remain to be fully elucidated, it is generally accepted that postsynaptic Ca₂₊ pulses play a fundamental role. A possible plasticity mechanism, called backpropagation-activated Ca₂₊ (BAC) firing, involves coincidence of strong proximal and distal inputs that may lead to dendritic spikes and bursts of axosomatic APs (Larkum et al., 1999; Hamilton et al., 2010). Synapses that cooperate on their quest to associate different inputs potentiate, whereas synapses that do not cooperate and/or do not succeed to establish an associational signal depress. BAC firing potentiates those synapses that cause it, thereby increasing the probability that this selected subset of synapses leads to BAC firing at the next time they are active (Larkum, 2013). Calcium-dependent dendritic spikes with attenuating amplitude (dCaAPs) and dendritic Na₊ spikes allow NMDAR dependent LTP in distal synapses (Gidon et al., 2020). Ca₂₊ pulses of short duration and high amplitude induce LTP (Evans and Blackwell, 2015). Low levels of Ca₂₊ lead to no changes in synaptic strength, medium levels cause LTD, and high levels lead to LTP (Lisman, 1989; Artola et al., 1990; Artola and Singer, 1993; Shouval et al., 2002; Graupner and Brunel, 2012). Potentiation can occur when an NMDA spike (also called a dendritic plateau potential) is generated. These plateau potentials provide a long and sufficiently high depolarization that leads to potentiation without generating postsynaptic action potentials. Plateau potentials can control plasticity at other synapses. Because they are more readily evoked in the terminal regions of basal dendrites, they undergo considerable attenuation and cause only subthreshold events at the soma. This long subthreshold plateau reduces the depolarization required to reach the spike threshold and, therefore, allows other weak inputs to reach the threshold. Active conductance in dendrites can normalize the efficacy of distal synapses and democratize dendrites by making distal and proximal synapses equally efficient in influencing somatic firing. Strong dendritic tapering attenuates electrotonic attenuation to such an extent that the dendritic tree becomes approximately isopotential (Otopalik et al., 2019). Therefore, dendritic structures may avoid attenuation. Inhibitory synapses on the dendritic tree prevent backward propagation of APs, while forward propagation remains intact (Wilmes et al., 2016; Iascone et al., 2020). Once a dendritic spike is generated, local inhibition can terminate it either transiently or permanently, depending on their relative timing. The classical “pre-/post-spike-timing-dependent-plasticity” (STDP) rule states that plasticity depends on pre-synaptic activity before and a sensed postsynaptic potential afterwards (Yang and Dani, 2014). STDP requires the generation of somatic Na₊ spikes and their backpropagation into the dendrites. Pair-based STDP cannot account for activity-dependent learning with weak inputs, which are not powerful enough to evoke bAPs. Finally, an increasing number of experimental studies have revealed plasticity mechanisms that do not rely on bAP but instead on local postsynaptic dendritic spikes or sub-threshold events for dendritic spikes (Ebner et al., 2019). Plasticity of distal feedback-associated synapses is a hot topic in studies exploring the idea of deep learning in the brain (Guerguiev et al., 2017; Richards et al., 2019).

A recent renaissance of dendritic computation has emerged through proposals of new model variants (Ostapoff et al., 1994; Voelker and Eliasmith, 2018; Payeur et al., 2019; Wybo et al., 2019; Gidon et al., 2020; Lepicard and Ann Piskorowski, 2020; Moldwin and Segev, 2020; Poirazi and Papoutsi, 2020; Takahashi et al., 2020; Banerjee et al., 2021; Callan et al., 2021; Martin et al., 2021; Moldwin et al., 2021; Stöckel and Eliasmith, 2021; Yang et al., 2021b). The octopus' cell acts as a synchronizer (McGinley et al., 2012). Functional simplicity is the overarching goal in constructing the dendritic tree with state-dependent temporal logic operators to sublimate many subtle details of neural morphology and rate kinetic dynamics. The tree is constructed simply by concatenating unit-base elements into unbalanced variable-length branches. The innervation patterns of the ANFs are given by the input activity matrix A(S_ij(t)). From the activated synapses, Dirac-like subthreshold voltage signals propagate down the dendrites to the soma, can collectively generate APs, and can produce their own unsupervised teaching signal by backpropagation into the dendrites. Kirchhoff's rules apply at the dendritic junctions, and attenuation and scattering of the signals are not considered. The current flows converge at the junctions and swell after the junctions. The soma is depolarized when the charge currents from the dendritic tree arrive synchronously and enough charge is accumulated to overcome the depolarization threshold. Backpropagating action potentials velocities are in the range of 226 μm/ms, which is 0.2–0.3 μm/μs (Senzai and Buzsáki, 2017) and 508 μm/ms for apical dendrites of layer five pyramidal neurons (basal dendrites, 341 μm/ms) (Nevian et al., 2007) and is similar to or lower than the estimated velocity in apical dendrites of other hippocampal principal neurons (Spruston et al., 1995b; Kim et al., 2012).

The condition that a delay trajectory in the receptive field of an octopus' cell leads to the same arrival time t_arrival at the soma for a vesicle in a dendrite branch i is given by:

Where t_i is the arrival time at the synapse connected to that branch and Δt_i the time the spike travels along the branch to the soma. Earlier arrival times are compensated by longer travel times in the partially shared dendrite pathways (Figure 2). The variation principle is to adjust all path lengths and velocities so that the formula is satisfied for each synapse, or at least for a subset of synapses.

The collectively triggered action potential travels along the axon and, simultaneously, a potential propagates backward from the soma into the dendrites until the most distant synapses are reached (Figure 2) (Brunner and Szabadics, 2016). The same paths defined by the delay elements (Ds) tree are taken on the backpropagation path. Attenuation and dispersion of the backpropagation signal flow are neglected in this work.

Learning at a synapse is given by a causal associative pre-post spike-timing dependent plasticity rule (STDP). A synapse is activated by presynaptic glutamatergic vesicles from the presynaptic ANF and enters an active sensitized state for a while (preconditioning). A triggered action potential at the soma generates a backward propagating signal into the dendrites (postconditioning signal) (Figure 2). The (pre, post) condition is satisfied when the synapse is in its active sensitized state and the backpropagating signal occurs before desensitization (Figure 2).

The weight update rule simplifies to:

Synapses consist of both AMPA and NMDA channels. The number of AMPARs is set to zero at the beginning. The maximum AMPAR conductance is chosen to be equal to the maximum NMDAR conductance. The weights of active synapses are updated when a somatic bAP is triggered. The weights of the synapses converge to a state of maximum conductance 1 or rest at the minimum state 0. The learning rate is adjustable and chosen such that one hundred backpropagation signals drive the synaptic weight w_j to its maximum 1.

To validate the proposed learning rule, we created a numerical model in the Python programming language (Python 3). The focus is on realizing a discrete-time model in an event-driven environment. The activation inputs of the octopus cells are given by the matrix A(S_ij(t)).

Stuart et al. (2016) give a detailed description of the morphology and functionality of dendrites in their compendium. According to this, the dendritic tree consists of directed graph elements (Mel et al., 2016) that are combined to form a more complex network. The soma is the root and the graph divides to reach all synaptic sites. Each node knows its child and predecessor nodes to realize conductive forward and backward propagation paths from synapses to the soma. The dendrites form a directed acyclic graph that is directed to either the soma or synaptic connections depending on the propagation state (forward/backward) (Figure 2). The dendritic tree is simplified to individual linearized dendrites in the form of tapped delay lines with a static delay of 18 μs per simulation time step. The dendritic delays are chosen to follow a logarithmic curvature given by the Greenwood formula (Greenwood, 1990). The delay values are assigned to each octopus' dendrites in a way that the octopus delays overlap over the receptive field (Figure 3).

The tokens of a trajectory travel to the soma, which accumulates all inputs over a short time window and generates an action potential when a certain threshold is crossed. When this event occurs, a wave is backpropagated, and each time the weight values of all activated synapses are increased by the value of the learning rate. The repeated backpropagation waves bring the synaptic weights to their saturation value of 1.0, putting the system in a steady state. The soma leakage is modeled by decreasing the membrane potential of the soma by a fixed decay value at each time step. The decay must be chosen very carefully because too large a value will cause the soma to see more than one trajectory and produce spikes at the output with a multiple of the central time interval, and too small a value would prevent the soma from firing at all.

To provide biologically plausible input for out numerical model an auditory front-end is used called SAM. SAM stands for Stimulation based on auditory modeling and it creates cochleagrams from a given prerecorded sound file as shown in Figure 4 for the vowel “a” sung by a woman on the note G2.

Our program package is designed for maximum flexibility to quickly create and study different topologies for pitch perception.

In the auditory model, 101 ANFs are arranged tonotopically along the frequency axis from the lowest to the highest frequency (Harczos, 2015). The spectral interval from C3 to G5 is examined using the presented model (Harczos and Klefenz, 2018). There are eleven octopus cells in this interval, each connected to nine separate ANFs. This wiring scheme represents a local receptive field for each octopus' cell. Adjacent receptive fields partially overlap and share multiple ANFs as inputs (Klefenz and Harczos, 2020).

An octopus' cell fires when a delay template segment matches a global trajectory and the corresponding synaptic connections are strengthened (Shamma and Dutta, 2019). Multiple octopus cells fire a series of spikes together when exposed to a common global trajectory and local receptive conditions are met. For quasi-stationary acoustic signals, nearly the same ANF trajectories occur repeatedly, and ANF firing patterns are nearly identical, as are the firing patterns of octopus cells. Two successive trajectories trigger firings of octopus cells at two different times in succession, forming inter-spike intervals (ISI). The entire process sequence is shown in a schematic diagram as an example in Figure 8.

Their time intervals are converted into a latency code for the first time by the transmission circuits between the octopus' cell and the ITN (Figure 9). The ITN fires first at the shortest time interval and last at the longest interval. Each intervening time interval is indicated by the corresponding FSL time. The collective task is to create, store, and interpret ISI histograms (Harpaz et al., 2021). The pitch neurons are connected to the corresponding bins. Each pitch neuron collects and counts the entries in its associated histogram bin. Inter-spike intervals are linearly proportional to FSL times. A start condition flag starts a clock counter at the FSL time of the corresponding minimum inter-spike interval (Verzi et al., 2018). Each pitch neuron fires when the entries in its FSL bins are above a fixed threshold number. The involved innervating synapses from ITNs to pitch neurons are increased by w_ij during the learning phase for this event. Collective learning of pitch by observing and interpreting associations of ICs by pitch neurons has been extensively studied in Klefenz and Harczos (2020).