Abstract

This letter examines the results of input-output (nonparametric) modeling based on the analysis of data generated by a mechanism-based (parametric) model of CA3-CA1 neuronal connections in the hippocampus. The motivation is to obtain biological insight into the interpretation of such input-output (Volterra-equivalent) models estimated from synthetic data. The insights obtained may be subsequently used to interpretat input-output models extracted from actual experimental data. Specifically, we found that a simplified parametric model may serve as a useful tool to study the signal transformations in the hippocampal CA3-CA1 regions. Input-output modeling of model-based synthetic data show that GABAergic interneurons are responsible for regulating neuronal excitation, controlling the precision of spike timing, and maintaining network oscillations, in a manner consistent with previous studies. The input-output model obtained from real data exhibits intriguing similarities with its synthetic-data counterpart, demonstrating the importance of a dynamic resonance in the system/model response around 2 Hz to 3 Hz. Using the input-output model from real data as a guide, we may be able to amend the parametric model by incorporating more mechanisms in order to yield better-matching input-output model. The approach we present can also be applied to the study of other neural systems and pathways.

1  Introduction

The hippocampus is one of the most important brain structures because it plays a critical role in episodic memory formation (Scoville & Milner, 1957; Tulving & Markowitsch, 1998), learning (Berry & Thompson, 1978; Whitlock, Heynen, Shuler, & Bear 2006), and spatial navigation (O'Keefe & Nadel, 1978; Moser, Moser, Forrest, Andersen, & Morris, 1995). The CA3 and CA1 regions of the hippocampus have been extensively studied because of the synaptic plasticity exhibited in their circuitry, such as the Schaffer collateral pathway (Collingridge, Kehl, & McLennan, 1983; Bliss & Collingridge, 1993). Another reason is that the CA3 and CA1 regions are vulnerable to many diseases (Small, Schobel, Buxton, Witter, & Barnes, 2011). For example, neuron loss in the CA3 and CA1 regions of the hippocampus is common in Alzheimer's disease (West, Coleman, Flood, & Troncoso, 1994; Padurariu, Ciobica, Mavroudis, Fotiou, & Baloyannis, 2012). Therefore, studying the signal transformations in CA3 and CA1 regions is crucial for understanding the function of the hippocampus and eventually building a neuroprosthesis to reconstruct the neural connections and restore impaired memory formation (Berger et al., 2001, 2012; Hampson et al., 2012, 2013).

The circuitry of the CA3 and CA1 regions is rather complex. As shown in Figure 1, the CA1 pyramidal cells receive excitatory inputs from the entorhinal cortex via the perforant pathway (PP) and the CA3 cells via the Schaffer collaterals (SC). The PP synapses are located at the distal apical dendrites, and the SC synapses are at the proximal dendrites (Otmakhova, Otmakhov, & Lisman, 2002; Andersen, 2007). Recent research indicates that some hippocampal GABAergic interneurons such as oriens lacunosum-moleculare (OLM) cells would facilitate the transmission from CA3 while reducing the influence of entorhinal cortex (Leão et al., 2012). In this letter, we consider only the signal transformations from the CA3 region to the CA1 region; the inputs from the entorhinal cortex are not included in our model. In the CA1 region, the principal cells receive not only excitatory glutamatergic inputs but also inhibitory GABAergic inputs, which originate mainly from interneurons (Freund & Buzsáki, 1996). Although the local GABAergic interneurons are a small proportion of the total number of neurons, they are responsible for regulating the excitation, controlling the precision of spike timing, and maintaining network oscillations (Sik, Penttonen, Ylinen, & Buzsáki, 1995; Fricker & Miles, 2001; Mann & Paulsen, 2007; Klausberger, 2009; Kullmann, 2011; Andersen, 2007). The interneurons are highly diverse and can be classified morphologically, neurochemically, and physiologically (McBain & Fisahn, 2001; Maccaferri & Lacaille, 2003). The CA1 pyramidal cells receive inhibitory GABAergic inputs from interneurons, such as basket, bistratified, axo-axonic, and oriens lacunosum-moleculare (OLM) cells. The basket and bistratified cells target the proximal apical dendrites, while the axo-axonic cells target the axons of CA1 cells. All of these cells take part in both feedforward and feedback inhibitory circuits (Sik et al., 1995; McMahon & Kauer, 1997; Cutsuridis, Cobb, & Graham, 2010; Müller & Remy, 2014). In this study, we ignore the axo-axonic interneurons, because 92% of the GABAergic synapses contact CA1 cells via their proximal dendrites (Megias, Emri, Freund, & Gulyas, 2001). The OLM interneurons, which act only in the feedback circuits, are found to inhibit the distal apical dendrites as well as disinhibit the proximal apical dendrites of CA1 cell (Leão et al., 2012).

Figure 1:

Schematic of the neural circuit of a CA1 cell in the hippocampus.

Figure 1:

Schematic of the neural circuit of a CA1 cell in the hippocampus.

Previously nonparametric input-output modeling methods, such as multiple-input multiple-output (MIMO) Volterra modeling, have been utilized to analyze the signal transformations between the CA3 and CA1 regions (Zanos et al., 2008; Sandler et al., 2015a, 2015b; Song et al., 2007, 2013, 2016). Although these methods provide valuable predictive models in a nonlinear dynamic context, they do not offer much insight into the underlying biological mechanisms. On the other hand, detailed compartmental models have been developed in an effort to represent the signal transformations within CA1 circuits (Cutsuridis, Cobb, & Graham, 2010; Olypher, Lytton, & Prinz, 2012). However, this task is daunting because of the difficulty in incorporating all the biological details of the diversity and complexity of synaptic plasticity (Malenka & Bear, 2004; Andersen, 2007), dendritic structure (Spruston, Schiller, Stuart, & Sakmann, 1995), neuronal backpropagation (Golding, Kath, & Spruston, 2001), and interneuron connections (Maccaferri & Lacaille, 2003). In this letter, we simplify the model of CA3-to-CA1 signal transformations to represent the essential mechanisms and the associated neural circuitry between CA3 and CA1, as shown in Figure 2. The simplified structural/parametric model consists of six modules, which we describe in section 2.

Figure 2:

Structure of simplified CA3-CA1 structural/parametric model.

Figure 2:

Structure of simplified CA3-CA1 structural/parametric model.

In order to evaluate the model with synthetic and experimental data, we use a newly developed data-based method, neuronal mode network (NMN), which can be used to analyze either synthetic data generated from simulations of the model in Figure 2 or actual experimental data. NMN is capable of extracting the dynamics and the static nonlinearities in neural systems under minimum assumptions. Comparing NMN results from synthetic data with NMN results from experimentally collected data in the human hippocampus, we can evaluate the simplified CA3-CA1 model of Figure 2 and gain insight into the origin of the dynamics and nonlinearities contained within the experimental data.

The organization of this letter is as follows. We first introduce the simplified structural/parametric CA3-CA1 model, which consists of six modules, in section 2. Then we give the basic descriptions of the nonparametric data-based modeling method NMN in section 3. The simulation results and analysis are shown in section 4. The discussion and conclusions are given in section 5.

2  Parametric Model of CA3-CA1 Signal Transformation

2.1  Schaffer Collateral Presynaptic Model

The presynaptic activities introduce the use-dependent plasticity such as short-term depression and facilitation within the timescale from milliseconds to seconds (Dittman, Kreitzer, & Regehr, 2000; Zucker & Regehr, 2002). This short-term presynaptic plasticity is critical for neural information processing, such as regulation of the circuit properties in SC (Pouille & Scanziani, 2004; Regehr, 2012) and the function as an adaptive filter for high-frequency discharges (Klyachko & Stevens, 2006; Kandaswamy, Deng, Stevens, & Klyachko, 2010). In order to model the short-term plasticity, we adopt the classic Dittman's model based on the residual calcium hypothesis (Dittman et al., 2000), which has also been used for comparison with nonparametric Volterra models (Song, Marmarelis, & Berger, 2009). The short-term plasticity of the is expressed as the presynaptic weight
formula
2.1
where represents the facilitation factor and is the depression factor, both ranging from 0 to 1. is described by the equations
formula
2.2
formula
2.3
formula
2.4
where is the baseline probability of transmitter release and is the concentration of the putative calcium-bound molecule, which decays with time constant after a surge of at time when an action potential occurs. is the Dirac delta function. is the affinity of . and follow the relation shown in equation 2.4, where is the facilitation ratio.
The depression factor is defined by
formula
2.5
formula
2.6
formula
2.7
where is the rate of recovery that is bounded by and . is the concentration of another putative calcium-bound molecule, which decays with time constant after a surge of at time when an action potential occurs. is the affinity of . We adopt the SC presynaptic parameters from Dittman et al. (2000), which are summarized in Table 1.
Table 1:
Parameters of the SC Presynaptic Model.
 2.2 
 0.24 
 100 ms 
 
 50 ms 
 0.03 ms 
 0.002 ms 
 
 
 2.2 
 0.24 
 100 ms 
 
 50 ms 
 0.03 ms 
 0.002 ms 
 
 

2.2  Schaffer Collateral Postsynaptic Model

After the neurotransmitters are released in the presynaptic sites, they diffuse across the synaptic cleft and reach the postsynaptic membrane, activating the respective synaptic channels and producing the postsynaptic currents. Glutamate is the main excitatory neurotransmitter in the SC pathway, where the AMPA receptor (AMPAr) and the NMDA receptor (NMDAr) are the main ionotropic receptors for glutamate (Dingledine, Borges, Bowie, & Traynelis, 1999). The AMPAr mediates the fast excitatory postsynaptic potentials (EPSPs), while the NMDAr mediates the slow EPSPs (Destexhe, Mainen, & Sejnowski, 1994). The nonlinear current-voltage ( relationship of the NMDAr reflects the blockage of at the resting potential, which is crucial for the coincidence detector since the activation of the NMDAr requires both the presynaptic glutamate release and the postsynaptic membrane depolarization (Tsien, Huerta, & Tonegawa, 1996). The dynamics of the AMPAr conductance can be expressed as the classic biexponential equation (Destexhe et al., 1994; Kleppe & Robinson, 1999; Dayan & Abbott, 2001):
formula
2.8
where is the maximum conductance of the AMPAr, and are the relevant time constants, and is the normalizing factor, which guarantees that the conductance would reach its maximum value .
The dynamics of the NMDAr conductance can also be expressed in the biexponential form (Kapur, Lytton, Ketchum, & Haberly, 1997; Wang, Song, & Berger, 2002):
formula
2.9
where is the maximum conductance of the NMDAr, and are the related time constants, and is the normalizing factor; is the concentration of , and is the dendritic membrane potential; and are constants related to the voltage-dependent blockage.
The EPSCs of the AMPAr and NMDAr can be expressed by the convolutional relations
formula
2.10
formula
2.11
where and are the numbers of AMPAr and NMDAr, respectively; is the presynaptic weight calculated from equation 2.1; and represents the convolution operator. and are the reverse potentials of the AMPAr and NMDAr, respectively. All the parameters of AMPAr and NMDAr are summarized in Table 2 (adapted from Wang et al., 2002).
Table 2:
Parameters of AMPAr and NMDAr.
 10 pS 
 1 ms 
 0.4 ms 
 0 mV 
 30 pS 
 55 ms 
 0.6 ms 
 0.33 mM 
 0.14 mV 
 0 mV 
 10 pS 
 1 ms 
 0.4 ms 
 0 mV 
 30 pS 
 55 ms 
 0.6 ms 
 0.33 mM 
 0.14 mV 
 0 mV 

2.3  Feedforward Inhibition

As shown in Figure 1, the feedforward interneurons of the CA3-to-CA1 circuit include bistratified, basket, and axo-axonic cells (Cutsuridis, Cobb, & Graham, 2010). To simplify our model, we treat only the feedforward interneurons as a module rather than model them separately. We also omit the interactions between interneurons such as the mutual inhibitions between bistratified and basket cells (Andersen, 2007; Cutsuridis, Cobb, & Graham, 2010; Saudargiene, Cobb, & Graham, 2015). The feedforward inhibition module in Figure 2 receives input from the CA3 cell and generates the IPSPs that are integrated by the CA1 dendrites. For the feedforward inhibitions, the plasticity of interneurons (Kullmann & Lamsa, 2007; Ali, 2011) is not considered in this model. The IPSPs generated by feedforward inhibition for GABA and GABA receptors are described by the convolutional relations,
formula
2.12
formula
2.13
where and are the numbers of GABA and GABA receptors for the feedforward inhibitions, is the time delay introduced by interneurons, and are the reverse potentials, and is the membrane potential of the dendrites. The conductances of the GABA and GABA receptors are expressed as the biexponential functions
formula
2.14
formula
2.15
where and are the maximum conductances; and are normalizing factors that guarantee that the conductances reach their maximum values; and , , , are time constants. The parameters of the GABA and GABA receptor are adopted from Cutsuridis, Graham, Cobb, and Vida (2010) and Gauck and Jaeger (2003), which are summarized in Table 3.
Table 3:
Parameters of GABAAr and GABABr.
 70 pS 
 8 ms 
 1 ms 
 70 mV 
 40 pS 
 100 ms 
 35 ms 
 70 mV 
 2 ms 
 70 pS 
 8 ms 
 1 ms 
 70 mV 
 40 pS 
 100 ms 
 35 ms 
 70 mV 
 2 ms 

2.4  Feedback Disinhibition

One of the most important types of feedback interneurons in the hippocampus is the OLM cell, which receives excitatory inputs from the axon collaterals of the CA1 cells (Cutsuridis, Cobb, & Graham, 2010; Saudargiene et al., 2015). The OLM interneurons are found to inhibit the distal apical dendrites and disinhibit the proximal apical dendrites of the CA1 cell. Therefore, the information flow from the SC pathway is enhanced, while the information flow from the entorhinal cortex is suppressed (Leão et al., 2012; Müller & Remy, 2014; Womelsdorf, Valiante, Sahin, Miller, & Tiesinga, 2014). As shown in Figure 1, the stratum raditaum and axo-axonic interneurons not only provide feedforward inhibitions but also participate in the feedback inhibitions. The effects of axo-axonic interneurons are excluded because of the relatively small number of GABAergic synapses on axons (Megias et al., 2001). In this module, we do not include the feedback inhibition effects caused by stratum raditaum interneurons because the feedforward inhibitions play a more important role (Paz & Huguenard, 2015; Ferrante & Ascoli, 2015). The inhibitory effects on the distal dendrites caused by OLM cells are also not included in the feedback module in Figure 2 because the amplitudes of IPSPs attenuate dramatically as the dendrites are located farther away from the soma (Spruston, 2008). Therefore, the feedback module only includes the OLM disinhibition effects on the proximal dendrites, which introduce the additional membrane potential depolarization (Leão et al., 2012). This depolarization caused by OLM cells is quantified by the equations:
formula
2.16
formula
2.17
formula
2.18
where and are the numbers of GABA and GABA receptors for the feedback disinhibitions, is the time delay introduced by interneurons, and is the resistance of the dendrites, which is chosen as 1 without loss of generality.

2.5  Synaptic Integration and the H-H Model

Each CA1 neuron receives numerous synaptic inputs from different locations, which spread at different distances from the soma. The resulting EPSPs and IPSPs from synapses are location dependent (Allam et al., 2015). In order to simplify the integration of the synaptic currents, we use a uniform synapse model rather than treat each synapse differently, as shown in sections 2.3 and 2.4. Therefore, the dendritic membrane potential in our model is treated as the integrated value rather than the membrane potential from a specific synapse. To model the dynamic processes for the generation of an action potential (AP) by a CA1 cell, we adopt the classic Hodgkin-Huxley (H-H) equations (Hodgkin & Huxley, 1952). The total current injected into the H-H model is
formula
2.19
The main H-H equation is
formula
2.20
where is the membrane potential and is the membrane capacitance. The maximum conductances of potassium sodium, and leakage ion channel are represented by , and . are the reverse potentials for each channel. We use the standard H-H parameters from the giant axon of a squid, which are summarized in Table 4. The gating variables , , are responsible for the voltage-dependent activation and deactivation of the channels, which are described by the first-order differential equations:
formula
2.21
formula
2.22
formula
2.23
formula
2.24
formula
2.25
formula
2.26
It should be noted that the resting potential is 0 mV; thus, we have to calibrate the resting potential by adding the offset voltage, so that the membrane potential of the soma/axon is
formula
2.27
where is chosen as 65 mV, which matches the experiments in vivo (Henze & Buzsaki, 2001).
Table 4:
Parameters of the H-H Equations.
 1 uF/cm 
 12 mV 
 115 mV 
 10.6 mV 
 36 mS/cm 
 120 mS/cm 
 0.3 mS/cm 
 1 uF/cm 
 12 mV 
 115 mV 
 10.6 mV 
 36 mS/cm 
 120 mS/cm 
 0.3 mS/cm 

2.6  Backpropagation of Action Potential

After the action potential (AP) is initiated in the axon of the CA1 cell, it will backpropagate to the dendritic branches (Stuart, Spruston, Sakmann, & Häusser, 1997). Backpropagation (BP) is hypothesized to provide the necessary depolarization to remove the Mg block of the NMDAr channels, which is essential for induction of long-term potentiation (Stuart et al., 1997; Johnston et al., 2003). Another effect of BP is to reset the synaptic integration by depolarizing the dendritic membrane potential close to the reverse potential of glutamate receptors (Stuart et al., 1997; Häusser, Major, & Stuart, 2001). One approach to model the BP is based on the cable theory and compartment models based on the realistic morphology of dendrites and axons (Larkum, Rioult, & Lüscher, 1996; Lüscher & Larkum, 1998). To simplify the BP model, we assume that the BP action potentials are produced by convolving the membrane potential from soma/axon with a low-pass filter, and the amplitude of the backpropagating AP is attenuated by a certain proportion,
formula
2.28
where is the shrinkage factor and is the time constant of BP. We choose to approximate the realistic BP effects (Stuart et al., 1997), as illustrated in Figure 3. It should be noted that the activity-dependent properties (Spruston et al., 1995) and dichotomous behavior of the backpropagating APs (Golding et al., 2001) in CA1 cells are not included in our BP module. The membrane potential of the dendrites in the CA1 cell is
formula
2.29
Figure 3:

Illustration of the simplified backpropagation model, where is the membrane potential of the soma/axon and is the backpropagating potential.

Figure 3:

Illustration of the simplified backpropagation model, where is the membrane potential of the soma/axon and is the backpropagating potential.

3  The Input-Output Model: Neuronal Model Network

In order to evaluate the mechanism-based structural model and compare it with the actual experimental data, we use a newly developed data-based method, the neuronal mode network (NMN), in order to analyze the data generated from the parametric model shown in Figure 2. The NMN method stems from the Volterra-equivalent network (Marmarelis & Zhao, 1997; Marmarelis, 2004; Geng & Marmarelis, 2015, 2017) and is capable of extracting the dynamics and the static nonlinearities in neural systems under minimum assumptions. As shown in Figure 4, the NMN follows the general framework of the classic cascade model (Marmarelis & Marmarelis, 1978; Dayan & Abbott, 2001; Marmarelis, 2004). Compared with the modeling method using principal dynamic modes (PDMs) (Marmarelis & Orme, 1993; Marmarelis, 1997; Marmarelis et al., 2013, 2014) and the widely used approach of generalized linear models (Paninski, 2004; Truccolo, Eden, Fellows, Donoghue, & Brown, 2005), the NMN would offer improved model estimation and prediction results.

Figure 4:

The architecture of (a) the traditional cascade model and (b) NMN. The NMs are linear filters and the MDM nonlinearly transforms the outputs of NMs into firing probabilities.

Figure 4:

The architecture of (a) the traditional cascade model and (b) NMN. The NMs are linear filters and the MDM nonlinearly transforms the outputs of NMs into firing probabilities.

As illustrated in Figure 4, the NMN consists of two cascaded stages, a set of near-optimal linear filters, the neuronal modes (NMs), and the multidimensional mapping (MDM) that nonlinearly transforms the NM output combinations into firing probabilities. The objective of NMN is to find a set of NMs in order to separate the combinations of NM output values that cause an output spike from combinations that do not. This separation between the two states of the output is described by the estimated MDM. The NMs of the system can be represented as the weighted sum of discrete Laguerre function (DLF) that serve as basis functions:
formula
3.1
formula
3.2
where is the Laguerre parameter, which controls the exponential declining rate of DLF. The outputs of the NMs can be expressed as
formula
3.3
where is the convolution between the input and the respective DLF:
formula
3.4
The MDM transforms the NM outputs into a firing probability by a multidimensional histogram according to Bayes' rule:
formula
3.5
Figure 5 is an illustration of how to construct the MDM. Let and be the outputs of the two NMs and use and to represent the NM output combinations that cause an output spike or not, respectively. The scatter plot of all and points is shown in Figure 5b. The scatter plot has pixels, and the number of and falling in each pixel is denoted as and , respectively. The firing probability corresponding to each pixel can be calculated as
formula
3.6
We use a “heat map” in Figure 5c to represent the MDM. The red areas correspond to high firing probability, which is termed the trigger region (TR). The blue areas correspond to low firing probability, which is termed the zero region (ZR). In this letter, due to the limited amount of human hippocampus data, we restrict the MDM to be two-dimensional. The details of choosing the cost function and training the NMN model are summarized in appendixes A and B.
Figure 5:

Illustration of MDM. (a) The outputs of and . The red points represent the combination of NM output values leading to output spikes . The unmarked points represent the combination of NM output values corresponding to zero output. (b) The scatter plot of the spiking points and the nonspiking points . (c) The constructed MDM after calculating the firing probability in each pixel.

Figure 5:

Illustration of MDM. (a) The outputs of and . The red points represent the combination of NM output values leading to output spikes . The unmarked points represent the combination of NM output values corresponding to zero output. (b) The scatter plot of the spiking points and the nonspiking points . (c) The constructed MDM after calculating the firing probability in each pixel.

4  Modeling Results

4.1  Regular Case

In order to generate the output response similar to the regular hippocampal CA1 cell, we need to set appropriate parameters for the number of glutamate and GABA receptors. The ratio of is chosen as 4:1 according to Hu, Bouteiller, Song, Baudry, and Berger (2015). Since the total number of glutamate receptors and GABA receptors is unknown in a CA1 cell, they should be carefully balanced in order to produce the naturally spiking activity, as shown in Table 5.

Table 5:
Numbers of Glutamate and GABA Receptor Channels.
  
  
  
  
  
  
  
  
  
  
  
  

The input is generated by the homogeneous Poisson model (Heeger, 2000), and the mean firing rate is 4.37 spikes/sec. The responses of each module in the system are shown in Figure 6. The mean firing rate of the output is 4.78 spikes/sec. As shown in Figure 6, the presynaptic short-term plasticity plays an important role in regulating the synaptic strength. For example, from 6800 ms to 6900 ms, two spikes in the CA3 cell fire very closely and cause a significant drop of synaptic strength, which prevents the CA1 cells from being overexcited. In the postsynaptic module, AMPAr and NMDAr mainly contribute to the excitatory currents, but they also contribute to the brief inhibitory currents when the dendritic membrane potential surpasses the reverse potential of AMPAr and NMDAr (0 mV), which would reset the excitatory currents after the AP backpropagates to the dendrites. The feedforward GABAAr inhibitory currents are fast and transient, which reduces the potential firing shortly after the generation of AP, while the GABABr inhibitory currents are slow and enduring, which reduces the firing after a relatively long time period.

Figure 6:

Simulation results of the CA3-CA1 parametric model. (a) The input spikes of the parametric model. (b) The response of SC presynaptic model. (c) The postsynaptic currents. (d) The backpropagation potential , feedback potential , and dendritic membrane potential . (e) The membrane potential of the dendrite/soma. (f) The output spikes of the parametric model.

Figure 6:

Simulation results of the CA3-CA1 parametric model. (a) The input spikes of the parametric model. (b) The response of SC presynaptic model. (c) The postsynaptic currents. (d) The backpropagation potential , feedback potential , and dendritic membrane potential . (e) The membrane potential of the dendrite/soma. (f) The output spikes of the parametric model.

Then we feed the input and output into the NMN model to obtain the system dynamic (NMs) and nonlinearity (MDM). Figure 7 shows the NMs and MDMs, which are obtained from the simulated data and from actual data collected in the human hippocampus during the intraoperative performance of a delayed match-to-sample task. The experiment details and the key NMN analysis results of the collected human hippocampus data are described in appendix C. Due to the limited amount of the data, it is likely that the SISO NMN analysis would bias toward certain frequency bands, and the estimated NMs and MDMs from the human hippocampus data exhibit high variability (shown in Figures 11 and 12). To mitigate this problem, we invoke the concept of global NMs (GNMs) that are obtained via singular value decomposition of the matrix composed of all estimated NMs of the functionally connected input-output neurons. The obtained GNMs of the human hippocampus data are shown in Figure 7 (red solid lines) and represent the common dynamic components of the MIMO neural system that serve as a functional coordinate basis for all neuronal dynamics in the subject system. After substituting the individual NMs with the global NMs, the recalculated MDMs become more regular, shown in Figure 12. The uniform and consistent elliptical shapes may indicate similar underlying mechanisms in the CA3 and CA1 neuronal connections. Therefore, we would like to examine if the simulation results obtained from the synthetic CA3-CA1 system would yield similar NMs and MDMs.

Figure 7:

(a) The obtained NMs from the simulated data (blue dashed lines) and real hippocampus data (red solid lines) in the time domain and frequency domain. (b) The obtained MDM from the simulated and the real hippocampus data.

Figure 7:

(a) The obtained NMs from the simulated data (blue dashed lines) and real hippocampus data (red solid lines) in the time domain and frequency domain. (b) The obtained MDM from the simulated and the real hippocampus data.

As shown in Figure 7, the NMs and MDMs obtained from the simulated and real data share some intriguing similarities. For example, NM #1 from simulated and real data exhibits a resonant peak around 3 Hz and a similar high-pass characteristic. The TRs and ZRs from both the simulated and real data are well separated with respect to the output of NM #1, although there is an additional ZR for the simulated data at higher values of the output of NM #1.

4.2  Deactivation of Feedback Disinhibition

Research studies have shown a significant loss of OLM cells in epilepsy (Cossart et al., 2001; Dinocourt, Petanjek, Freund, Ben-Ari, & Esclapez, 2003). In order to explore how the deactivation of the feedback disinhibition caused by OLM cells loss would affect the output activity of the CA1 cell, as well as the dynamics and nonlinearities of the system, we compute the model response after deactivation of the feedback inhibition module, as shown in Figures 8a to 8d. The output of the CA1 cell shows intermittent bursting with a firing rate of 10.74 spikes/sec. The feedback depolarization potential is 0 mV, and therefore, the dendritic membrane potential is always below the reverse potential of NMDA and AMPA, so that the NMDAr and AMPAr contribute only to the excitatory currents. Although the feedback disinhibition seems to make the CA1 cell excitatory, the overall effect turns out to be inhibitory. In a comparision of Figure 6c and Figure 8a, the feedback disinhibition would cause the additional depolarizations of the dendrites, push the membrane potential to surpass the reverse potential of NMDAr and AMPAr (0 mV), and cause the transient inhibitory currents. According to Figure 6c, NMDAr plays a dominant role in the transient inhibitory currents because of the nonlinear I-V relationship and slow dynamics of the conductance. The obtained NMs and MDM are shown in Figures 8e and 8f. The obtained NM #1 acts as a low-pass filter, and the NM #2 has a resonance peak around 1 Hz. There are no resonance peaks of the NMs at higher frequencies in this case. Although the obtained TR of the MDM is clearly clustered, it is surrounded by the ZR rather than being well separated as in Figure 7b.

Figure 8:

Simulation results of the CA3-CA1 parametric model when the feedback disinhibition module is disabled. (a) The postsynaptic currents. (b) The backpropagation potential , feedback potential , and dendritic membrane potential . (c) The membrane potential of the dendrite/soma. (d) The output spikes of the parametric model. (e) The obtained NMs. (f) The obtained MDM.

Figure 8:

Simulation results of the CA3-CA1 parametric model when the feedback disinhibition module is disabled. (a) The postsynaptic currents. (b) The backpropagation potential , feedback potential , and dendritic membrane potential . (c) The membrane potential of the dendrite/soma. (d) The output spikes of the parametric model. (e) The obtained NMs. (f) The obtained MDM.

4.3  Deactivation of Feedforward Inhibition

It has been observed that the feedforward inhibition is lost in epilepsy (Paz & Huguenard 2015), so it is important to explore how the removal of feedforward inhibition would affect the behavior of the model. When we remove the feedforward inhibition module in Figure 2 and keep the rest of the parameters the same as section 4.1, the system exhibits sustained bursting output of a very high firing rate of 94.00 spikes/sec, as shown in Figure 9. The feedback disinhibition depolarizes the membrane potential of dendrites and helps to remove the Mg blockage of NMDAr and generate more excitatory currents that facilitate the generation of AP. The disinhibition also helps to elevate the dendritic membrane potential to surpass the reverse potential of the glutamate receptors (0 mV) that generate the inhibitory currents. However, the excitatory effects are stronger than the inhibitory effects, and the overall outputs exhibit sustained bursting of spikes. The obtained NMs shown in Figure 9 are elongated and exhibit resonant peaks around 1 Hz, while higher-frequency resonances are absent. The obtained MDM is blurred, and the TR cannot be clearly distinguished from the ZR.

Figure 9:

The simulation results of the CA3-CA1 parametric model when the feedforward inhibition module is disabled. (a) The postsynaptic currents. (b) The backpropagation potential , feedback potential , and dendritic membrane potential . (c) The membrane potential of the dendrite/soma. (d) The output spikes of the parametric model. (e) The obtained NMs. (f) The obtained MDM.

Figure 9:

The simulation results of the CA3-CA1 parametric model when the feedforward inhibition module is disabled. (a) The postsynaptic currents. (b) The backpropagation potential , feedback potential , and dendritic membrane potential . (c) The membrane potential of the dendrite/soma. (d) The output spikes of the parametric model. (e) The obtained NMs. (f) The obtained MDM.

5  Discussion and Conclusion

In this letter, we have introduced a simplified mechanism-based CA3-to-CA1 parametric model (shown in Figure 2) that captures the main neuronal connections and mechanisms that contribute to the signal transformations from the CA3 to the CA1 region in the hippocampus. We have examined each module and provided the underpinnings of the mathematical formulation of the respective parametric model in section 2. In order to evaluate the resulting mechanism-based parametric/structural model, we use a newly developed databased method of input-output modeling, the neuronal mode network (NMN), to obtain an input-output model from the synthetic data generated from the parametric model. The results indicate that the simplified mechanism-based CA3-to-CA1 model exhibits similar input-output system dynamics and nonlinearities as observed using real data collected in the human hippocampus (see Figure 7). Specifically, the obtained MDMs indicate that the NM #1 (with a resonant peak around 2–3 Hz for both synthetic and real data) is mainly responsible for spike generation at the CA1 neuron. However, there are also some discrepancies observed between the input-output (MNM) models obtained from synthetic and real data, which may be caused by the exclusion of inputs from the entorhinal cortex or possible oversimplifications in the parametric model.

The loss of GABAergic interneurons and the dysfunction of hippocampal neural circuits are considered to be responsible for epilepsy (Cossart et al., 2001; Dinocourt et al., 2003; Ang, Carlson, & Coulter, 2006; Ben-Ari, Crepel, & Represa, 2008; Goldberg & Coulter, 2013; Paz & Huguenard, 2015). Thus, it is useful to explore the effects of feedforward inhibition and feedback disinhibition, as discussed in sections 4.2 and 4.3. The results shown in Figures 8 and 9 indicate that the deactivation of either feedforward inhibition or feedback disinhibition causes seizure-like bursting of action potentials (APs). When we compare the results shown in Figures 7 and 8, we can see that the feedback disinhibition effects are not merely excitatory or inhibitory. On one hand, the disinhibition helps the NMDAr to further depolarize in order to remove the Mg blockage and generate more excitatory currents that facilitate the generation of AP. On the other hand, the disinhibition helps to reduce the bursting of APs by elevating the dendritic membrane potential to surpass the reverse potential of the glutamate receptors (0 mV) that would reset the excitatory currents after the AP backpropagates to the dendrites.

When we compare the results from Figures 7 and 9, we can see that the feedforward inhibition effects are essential to prevent the overbursting activity of the CA1 cell. Using the NMN as the tool to analyze the data, we found that the neural dynamics concentrated on the very low-frequency bands (1 Hz) and higher-frequency dynamics are absent in both cases, which indicate that the feedforward inhibition together with feedback disinhibition are essential for modulating the oscillations in the CA3-to-CA1 signal transformation process. Comparing Figures 8f and 9f with 7a, we can see that another role of the GABAergic interneurons is to maintain the separation of the TR and ZR in the MDM, which is critical in order to preserve the spiking timing precision.

In summary, the simplified parametric model shown in Figure 2 may serve as a useful tool to study the signal transformations in the hippocampal CA3-CA1 regions. The results in Figures 8e and 9e indicate that deactivation of the feedback disinhibition module and feedforward inhibition module would lead to different dynamics/resonant frequencies of the system compared with the actual data in Figure 7a. In addition, Figures 8c and 8d, and 9c and 9d, suggest that significant loss of interneurons would cause the epileptic activities, since the GABAergic interneurons are responsible for regulating neuronal excitation, controlling the precision of spike timing, and maintaining network oscillations, in a manner consistent with previous research studies (Sik et al., 1995; Fricker & Miles, 2001; Mann & Paulsen, 2007; Klausberger, 2009; Kullmann, 2011; Andersen, 2007). The input-output models obtained from simulated/synthetic and real data exhibit similarities in demonstrating the importance of a dynamic resonance in the system/model response around 2 Hz to 3 Hz (see Figure 7). Recent studies also have shown that the “slow-theta” in the human hippocampus (around 3 Hz) is evident when the subjects are performing the episodic memory task and virtual navigation task (Lega, Jacobs, & Kahana, 2012; Watrous, Fried, & Ekstrom, 2011; Watrous et al., 2013; Jacobs, 2014). However, the neural rhythms of slow-theta oscillations observed in EEG recordings or local field potentials are collected activities of a large population of neurons rather than the single pair of CA3-CA1 neurons. One limitation of our current model is that it focuses on the signal transformation of a single pair of CA3-CA1 neurons, which may not able to represent all the dynamics of the hippocampal systems. In the future, a population model needs to be further implemented to explore the genesis of oscillations in the hippocampus, especially the theta-gamma coupling relationships (Kopell, Börgers, Pervouchine, Malerba, & Tort, 2010; Lisman & Jensen, 2013). In addition, the approach could be extended to the study of the dynamics of the hippocampal neural network in well-characterized conditions, such as a pair of place cells (O'Keefe & Burgess, 1996) or concept cells (Quiroga, 2012).

Appendix A:  Cost Function of NMN

The objective of training the NMN is to find a set of near-optimal NMs that separate the spiking points from the nonspiking points in the MDM. The parameters we need to optimize are and , which determine the shapes and time span of NMs. The cost function should be carefully selected to quantize the separation between the spiking and nonspiking points. We propose a cost function adapted from the Ising model (Ising, 1925) and Potts model (Wu, 1982), which is defined as
formula
A.1
where is the normalization factor and is the firing probability at the th pixel, which is defined in equation 3.6. The notation represents the bonds formed by neighboring pixels and , which is illustrated in Figure 3. This Ising energy cost function encourages the clustering of spiking points to form TR, because the cost function is reduced if both and are large (see Figure 10). In addition, it discourages the mix between the spiking and nonspiking points. Since the number of spikes and zeros are highly imbalanced, if the spiking and nonspiking points are well mixed, then all the pixels in MDM would have either zero or a very small firing probability, and there is no evident TR formed in MDM. The corresponding Ising energy will be high since . Thus, by minimizing this Ising cost function, the separation between the spiking and nonspiking points is optimized. However, if the input and output have no causal relationship, it is impossible to find a set of optimal NMs that can lead to such separation. In that case, the MDM after optimization would still be largely blurred, there is no evident TR, and the cost function would be around zero. Therefore, the MDM and its associated Ising energy could serve as an indicator of how strongly the input and output are functionally connected. Another important feature of this cost function is that it mitigates the imbalanced class problem of spike train data. The NMN method seeks to find a set of NMs that yield an optimal structure of the MDM instead of directly maximizing the prediction performance (as traditional learning algorithms do) that would suffer from the imbalanced class problem.
Figure 10:

Illustration of the Ising energy. Each dot represents a pixel in MDM. The red dot has eight nearest neighbors, which are represented by the black dots. The edge between the and one of the neighbors forms an energy bond (green line). The total energy is calculated by summing up all the energy bonds in MDM.

Figure 10:

Illustration of the Ising energy. Each dot represents a pixel in MDM. The red dot has eight nearest neighbors, which are represented by the black dots. The edge between the and one of the neighbors forms an energy bond (green line). The total energy is calculated by summing up all the energy bonds in MDM.

Appendix B:  Training NMN with Simulated Annealing Algorithm

Since the cost function is not differentiable with respect to the parameters that need to be optimized, traditional gradient descent methods like backpropagation cannot be used. Therefore, we propose the use of the simulated annealing (SA) algorithm (Kirkpatrick, Gelatt, & Vecchi, 1983), a global optimization method that does not require gradient information. The SA algorithm has been shown to be suitable for training Volterra-equivalent networks (Geng & Marmarelis, 2017). The following are the basic steps of SA to optimize the parameter set :

  1. Randomly initialize the parameter set , and set the temperature at a sufficiently large value . In our case, has to be in the range (0, 1) and the weights set within the range (1, 1) .

  2. Perform a small random move based on the current parameter set, so that the new one is . In this move, a single parameter is randomly chosen from the whole set following a uniform distribution, so that each time, every parameter would have the same probability of being selected. Then, is updated by the following rule,
    formula
    where the constant is the step size and is a random number drawn from a uniform distribution within the range [0, 1].
  3. Compute the cost functions and , respectively, to get the difference: .

  4. Calculate the acceptance probability:
    formula
    that is, if , we accept the new parameter set ; otherwise, is accepted with the probability defined above.
  5. Repeat steps 2 to 4 until it reaches an equilibrium, which may be close to a local minimum. In practice, we often choose a large but fixed number of iterations.

  6. Decrease the temperature according to the exponential decay cooling schedule (Nourani & Andresen 1998), where in in the range (0.95, 0.99):
    formula

When the Laguerre parameter is updated, the outputs of the DLF filter bank are recomputed (see equations 3.2 and 3.4). Since each time is updated only by decreasing or increasing by a step size and is within the range , there are only possible values. Therefore, we can calculate all the possible values of for each and store them in a look-up table to reduce the overall computational burden.

If we scale each NM weight with a constant and scale the axis of MDM with , then the overall system still remains the same. Since SA is a stochastic process, the final model trained by SA may have a very large constant , which will require numerous unnecessary iterations. To avoid this problem, we normalize the weights during the training process as
formula
B.4
so that
formula
B.5
With this normalization, the efficiency of the training process is largely improved. It should be mentioned that the estimation results are not sensitive to SA-related parameters, and SA can provide only near-optimal results (Geng & Marmarelis, 2017). The parameters of the SA training process used in this letter are summarized in Table 6.
Table 6:
Simulated Annealing Parameters.
SymbolNameValue
 Initial temperature 100 
 Cooling constant 0.99 
 Number of temperature droppings 1500 
 Number of iterations at each temperature 200 
 Step size 0.01 
SymbolNameValue
 Initial temperature 100 
 Cooling constant 0.99 
 Number of temperature droppings 1500 
 Number of iterations at each temperature 200 
 Step size 0.01 

Appendix C:  Human Hippocampus Data

Multineuronal data were recorded from the CA3 and CA1 regions of the human hippocampus intraoperatively in medically refractory focal epilepsy patients at the Wake Forest Baptist Medical Center. All procedures were reviewed and approved by the Institutional Review Board of Wake Forest University, in accordance with the National Institutes of Health. Patients underwent preoperative imaging to determine intraoperative electrode placement. The FDA-approved electrodes are capable of single-unit neuronal recording and field potential recording (Ad-Tech Medical Instrumentation Corporation, Racine, WI). Intraoperative placement of the depth electrodes was performed using either a stereotactic head frame (CRW Precision Arc, Integra Life Sciences, Plainsboro, NJ) or frameless stereotactic system (VarioGuide, Brainlab AG, Feldkirchen, Germany) at the discretion of the operating surgeon. Hippocampal electrode placement was targeted to penetrate the head of the hippocampus perpendicular to its long axis in order to record neuronal activity within the CA3 and CA1 subfields. The data are recorded intraoperatively during the delayed match-to-sample (DMS) task. The DMS trials were initiated by the examiner who manually triggered the presentation of a nonverbalizable image, constituting the sample presentation (SP) phase of the task. Completion of the SP task required the patient to touch the presented image, designated as the sample response (SR) phase.

The SR phase then initiated the delay interval phase of the trial during which the screen was blank for 5 to 15 seconds, randomly determined on a trial-by-trial basis. The delay interval was followed by the match presentation (MP) phase of the task, which consisted of the simultaneous display of two to seven images displayed at randomly selected spatial locations on the screen, one of which was the sample image. Touching either the previously presented image or another image was designated a match response (MR). Correct responses produced a reward tone followed by a blank screen. Incorrect responses caused by touching one of the nonmatch (distractor) images constituted a nonmatch response led to an error tone followed by a blank screen. Trials were separated by 5 seconds prior to initiation of the next trial. Single-unit neural activities from 10 patients were recorded and isolated using the Blackrock Cervello Elite electrophysiological recording system with a raw data acquisition frequency at 30,000 samples/sec without filtering. We set the frequency of the automatic spike-sorting algorithm at 30,000 samples/sec with 500 to 5000 Hz bandpass filtering.

We present the results of NMN analysis of data only from patient No. 2 as a pilot study. For patient No. 2, 14 trials were recorded with the correct DMS response and 22 trials with the incorrect response. Spike train data were recorded from seven CA3 neurons and eight CA1 neurons during the DMS task.

We analyzed the data during the 7 sec period after SP, with a bin size of 5 ms. We applied the NMN approach to determine the functional connectivity between the CA3 neurons as inputs and the CA1 neurons as outputs. Fifty-six neuronal input-output pairs were analyzed. Because of the observed variability of the estimated NMs and MDMs in Figure 11, we invoke the concept of global NMs (GNMs) that are obtained via singular value decomposition of the matrix composed of all estimated first or second NMs of the functionally connected input-output neurons, which is akin to the concept of global PDMs (Marmarelis et al., 2013, 2014). The obtained first and second GNMs are shown in Figure 7a and represent the common dynamic components of the MIMO neural system that serves as a functional coordinate basis for all neuronal dynamics in the subject system. We see that the first GNM peaks at around 3 Hz and 15 Hz, while the second GNM exhibits a low-pass characteristic. The new MDMs that are computed based on the obtained GNMs are shown in Figure 12 for the correct DMS responses. We can see that the TRs and ZRs are mainly separated by the first NM output, so this NM seems to be the important one for these neuronal pairs. The uniform and consistent elliptical shapes may indicate similar underlying mechanisms in the CA3 and CA1 neuronal connections.

Figure 11:

(a) The NMs and estimated from NMN for the human hippocampus training data. The unit of -axis is in seconds. The blue curves represent the NM 1 and red curves represent the NM 2. (b)The corresponding estimated MDMs, where J represents the energy function of NMN in equation A.1.

Figure 11:

(a) The NMs and estimated from NMN for the human hippocampus training data. The unit of -axis is in seconds. The blue curves represent the NM 1 and red curves represent the NM 2. (b)The corresponding estimated MDMs, where J represents the energy function of NMN in equation A.1.

Figure 12:

The new MDMs computed based on the GNMs of hippocampus CA3-to-CA1 data.

Figure 12:

The new MDMs computed based on the GNMs of hippocampus CA3-to-CA1 data.

Acknowledgments

This work was supported by NIH-NIBIB grant P41-EB001978 and DARPA contract N66601-09-C-2081 to the Biomedical Simulations Resource at University of Southern California. We thank the anonymous reviewers for their constructive comments and suggestions that greatly contributed to improving the letter.

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