The authors have declared that no competing interests exist.
Conceived and designed the experiments: JH DD. Performed the experiments: JH. Analyzed the data: JH LH. Contributed reagents/materials/analysis tools: JH LH DD. Wrote the paper: JH LH DD.
The prefrontal cortex is centrally involved in a wide range of cognitive functions and their impairment in psychiatric disorders. Yet, the computational principles that govern the dynamics of prefrontal neural networks, and link their physiological, biochemical and anatomical properties to cognitive functions, are not well understood. Computational models can help to bridge the gap between these different levels of description, provided they are sufficiently constrained by experimental data and capable of predicting key properties of the intact cortex. Here, we present a detailed network model of the prefrontal cortex, based on a simple computationally efficient single neuron model (simpAdEx), with all parameters derived from
Computational network models are an important tool for linking physiological and neuro-dynamical processes to cognition. However, harvesting network models for this purpose may less depend on how much biophysical detail is included, but more on how well the model can capture the
The prefrontal cortex (PFC) is a key structure in higher-level cognitive functions, including working memory, rule and concept representation and behavioral flexibility [
Computational network models are a highly valuable tool for driving forward such an understanding, as data from many different levels of experimental analysis can be integrated into a coherent picture. With respect to psychiatric conditions, it is of particular importance that models incorporate sufficient biological detail and exhibit physiological validity in order to serve as explanatory tools. Psychiatric conditions like schizophrenia are characterized by a multitude of abnormalities in diverse cellular and synaptic properties, transmitter systems, and neuromodulatory input [
Ultimately, the physiological validity of a computational model ought to be reflected in the degree to which it can reproduce and predict detailed aspects of the neural activity observed
In this work, we present a computational network model of the PFC which has high physiological validity and predictivity both at the single-neuron- (
The network model introduced in Materials and Methods aims to combine computational tractability with physiological validity. This balance is achieved by embedding a simple, reduced two-dimensional single neuron model into a realistic network architecture that is derived from the experimental literature. All model parameters were directly estimated from our own
At the single-cell level, the network is based on an approximation (simpAdEx [
(A) Example of the initial (upper curve) and steady-state (lower curve) input-output relation (f-I curve) of a single neuron. Black and gray curves show experimental data, red and blue curves indicate the simpAdEx model fits. (B) Voltage trace from a slice recording of a prefrontal cortical layer 5 pyramidal cell (black) and from the corresponding model cell (red) in response to the same fluctuating input current. The same neuron model and parameters as in Panel A were used [
parameter | PC L2/3 | FS | BT | MC | PC L5 |
---|---|---|---|---|---|
164.96 (59.11) | 59.58 (10.59) | 79.36 (14.83) | 81.12 (28.96) | 251.81 (82.61) | |
7.04 (1.72) | 5.34 (0.91) | 3.99 (0.51) | 2.98 (0.55) | 7.62 (2.09) | |
-85.00 (5.40) | -85.15 (5.81) | -84.63 (4.71) | -72.20 (7.64) | -80.57 (6.71) | |
Δ |
21.44 (6.42) | 19.58 (8.50) | 19.02 (4.08) | 22.30 (10.44) | 24.47 (5.96) |
121.78 (41.19) | 15.15 (2.71) | 43.56 (21.89) | 60.13 (15.05) | 107.48 (64.08) | |
7.29 (6.80) | 34.87 (37.88) | 6.65 (7.19) | 5.37 (5.78) | 8.27 (12.66) | |
-118.20 (38.14) | -90.16 (15.16) | -152.67 (49.15) | -55.89 (9.65) | -69.98 (14.45) | |
-52.40 (5.43) | -58.79 (9.82) | -59.95 (4.67) | -38.01 (6.03) | -48.69 (7.18) | |
-45.91 (7.22) | -51.01 (5.59) | -55.46 (4.39) | -36.94 (2.55) | -44.12 (7.28) |
Mean and standard deviation of the parameters of the simpAdEx model for the five different neuron types used in the network (PC: Pyramidal cell, FS: Fast-spiking interneuron, BT: Bitufted interneuron, MC: Martinotti cell)
Anatomically, the network is divided into two laminar components, representing the superficial layers L2/3 and deep layer L5 (
(A) Laminar structure of a single network column. Arrow widths represent relative strength of connections (black: excitatory, gray: inhibitory), i.e. the product of connection probability and synaptic peak conductance. (B) Left panel: Distribution of three different short-term plasticity types over different combinations of pre- and postsynaptic neuron types. Arrows from or to one of the shaded blocks (rather than from or to a single neuron type) denote connection types that are identical for all excitatory (PC) or inhibitory (IN) neurons. Where all three types are drawn, they are randomly distributed over all synapses between these two neuron types according to the probabilities given in the figure. Right panel: Illustration of the postsynaptic potential in response to a series of presynaptic spikes for three types of short-term synaptic plasticity for excitatory (E1 to E3) and inhibitory synapses (I1 to I3).
Layer | PC | IN-L | IN-CL | IN-CC | IN-F |
---|---|---|---|---|---|
47% | 3.1% | 2.6% | 2.6% | 2.1% | |
38% | 0.5% | 0.5% | 1.8% | 1.8% |
Relative numbers of cells for each type. PC: pyramidal cell, IN: interneuron, see
Neurons are connected by conductance-based synapses (AMPA, GABAA and NMDA) with kinetics estimated from electrophysiological data, short-term synaptic plasticity [
pre | post | |||
---|---|---|---|---|
PC L2/3 | PC L2/3 | 0.139 | 0.84 (0.49) | 1.55 (0.31) |
PC L2/3 | PC L5 | 0.233 | 0.95 (0.39) | 1.91 (0.17) |
PC L5 | PC L2/3 | 0.045 | 0.84 (0.28) | 2.75 (0.18) |
PC L5 | PC L5 | 0.081 | 0.88 (0.67) | 1.56 (0.44) |
PC L2/3 | IN-L L2/3 | 0.325 | 1.34 (1.09) | 0.96 (0.25) |
PC L2/3 | IN-CL L2/3 | 0.159 | 0.47 (0.20) | 0.96 (0.25) |
PC L2/3 | IN-F L2/3 | 0.290 | 0.25 (0.20) | 0.96 (0.25) |
PC L2/3 | IN-L L5 | 0.087 | 0.77 (0.86) | 1.18 (0.13) |
PC L2/3 | IN-CL L5 | 0.080 | 0.27 (0.16) | 1.18 (0.13) |
PC L2/3 | IN-F L5 | 0.150 | 0.14 (0.16) | 1.18 (0.13) |
PC L5 | IN-K L2/3 | 0.188 | 1.52 (0.63) | 1.05 (0.08) |
PC L5 | IN-CL L2/3 | 0.092 | 0.53 (0.12) | 1.05 (0.08) |
PC L5 | IN-F L2/3 | 0.168 | 0.28 (0.12) | 1.05 (0.08) |
PC L5 | IN-L L5 | 0.333 | 1.74 (1.12) | 0.60 (0.20) |
PC L5 | IN-CL L5 | 0.080 | 0.88 (0.70) | 0.60 (0.20) |
PC L5 | IN-F L5 | 0.362 | 0.28 (0.30) | 0.60 (0.20) |
IN-L L2/3 | PC L2/3 | 0.466 | 2.30 (1.98) | 1.25 (0.18) |
IN-CL L2/3 | PC L2/3 | 0.301 | 0.13 (0.48) | 1.25 (0.18) |
IN-F L2/3 | PC L2/3 | 0.710 | 1.91 (3.83) | 1.25 (0.18) |
IN-L L2/3 | PC L5 | 0.217 | 1.07 (0.92) | 1.54 (0.10) |
IN-CL L2/3 | PC L5 | 0.140 | 0.06 (0.22) | 1.54 (0.10) |
IN-F L2/3 | PC L5 | 0.330 | 0.89 (1.78) | 1.54 (0.10) |
IN-L L5 | PC L2/3 | 0.039 | 0.10 (0.01) | 1.44 (0.04) |
IN-CL L5 | PC L2/3 | 0.027 | 0.04 (0.01) | 1.44 (0.04) |
IN-F L5 | PC L2/3 | 0.040 | 0.07 (0.06) | 1.44 (0.04) |
IN-L L5 | PC L5 | 0.274 | 0.69 (0.10) | 0.82 (0.09) |
IN-CL L5 | PC L5 | 0.173 | 0.30 (0.05) | 0.82 (0.09) |
IN-F L5 | PC L5 | 0.282 | 0.50 (0.40) | 0.82 (0.09) |
IN L2/3 | IN L2/3 | 0.250 | 1.35 (0.35) | 1.10 (0.40) |
IN L5 | IN L5 | 0.600 | 1.35 (0.35) | 1.11 (0.40) |
Mean and standard deviation of the parameters of the synapses connecting the different pre- and postsynaptic neuron types (
Wherever possible, we used data from the rodent prefrontal cortex, or at least agranular cortices such as the motor cortex, which in rodents shows a similar layered anatomy as the PFC. Apart from the missing granular layer 4, specific features of the rodent PFC that are modeled here include an increased fraction of reciprocal compared to unidirectional connections [
To assess whether the network model can reproduce the dynamics of real prefrontal neurons
(A) Comparison of relative frequency histograms for three different spike time statistics between recordings from an
The
Without further tuning of network parameters beyond their derivation from slice-physiological and anatomical data, all these
While neurons firing at very low rates may go undetected using extracellular single-unit recordings, recording techniques that are less biased toward spiking neurons, such as calcium imaging or
(A) Estimated distribution of the standard deviation of the membrane potential from anaesthetized rats (gray) and simulated neurons (black) with non-zero firing rates. (B) Power spectrum of the local field potential obtained from experiments (gray) and simulations (black). The dotted lines illustrate the three power laws. The shaded region represents the mean ± the SEM at each point of the experimental distribution, as in
The local field potential (LFP) in the model was estimated as the sum of all synaptic currents (allowing excitatory and inhibitory currents to partially cancel). This is a reasonable approximation to the standard model of the LFP [
(A) Raster plot of the spike times in the network in response to an external input (gray line) to 10% of the L2/3 pyramidal cells. The input currents are
The transmission of transient stimuli between layers crucially depends on the heterogeneity of the neuronal parameters. With a 80% reduction in the variance of all parameter distributions (but no change in the means), the stimulus only elicits a response in L2/3, but is not transmitted to the output layer L5 anymore (
To further examine the transmission dynamics, we reproduced an
In the previous section we showed that the model can reproduce a wide range of characteristics of neural activity
(A) Maximum of the Kolmogorov-Smirnov test statistic (
The ratio of inputs into the two layers,
To estimate which range of
(A) Synaptic input current as a function of the number of columns. Shown are the averaged values over different neuron densities (mean ± SEM) as a function of column number for the inputs into L2/3 pyramidal cells (solid blue), L2/3 interneurons (solid red), L5 pyramidal cells (dotted blue) and L5 interneurons (dotted red). The region of currents which yield
The GABAA reversal potential
(A) Maximum of the Kolmorov-Smirnov test statistics (
Apart from the mean, we also analyzed how the distribution of the synaptic peak conductances affected
We presented a model of the prefrontal cortex which is entirely defined by electrophysiological and anatomical data, and is capable of reproducing a wide range of
The current model has a strong focus on its tight connection to data. Many existing network models of the neocortex are based on neurobiological findings as well [
An important simplification made in the present model is the reduction to two laminar components, leaving out layer 4 and 6 as well as the long-range fiber bundles and interneurons in layer 1. While layer 4 is missing in rodent PFC, layer 6 is only weakly connected to the other layers in our reference connectivity maps, which are based on the motor cortex [
The model exhibits a low fraction of spiking neurons, consistent with results from recording methods such as calcium imaging, which are not biased towards high firing rates (“dark matter theory” of neuroscience [
There are two main determinants of the high-fluctuation regime of the model: First, variability in the membrane potential requires variability in the synaptic parameters and in particular, the fat tail of the log-normal distribution of the synaptic weights. Second, the range between the firing threshold
Using the multivariate distributions of neuron parameters obtained from our
Thus, apparently quite subtle changes in the distributional properties of synaptic and cellular parameters (not affecting their means) may lead to major changes in network dynamics and functional connectivity among columns or areas, effects that have been proposed to underlie major psychiatric conditions like schizophrenia [
By varying the total input from a virtual population designed according to the same principles as the actually simulated network, we provided evidence that a larger network than the one actually simulated with anatomically realistic neuron densities should be capable of self-sustaining
Interestingly, the currents produced by this procedure are much higher in L5 compared to L2/3 (
In terms of space, input from just a few columns is sufficient to drive the network, as connectivity rapidly decays over the cortical extent. Nevertheless, a single column is not sufficient for driving the network because of the higher fraction of excitatory synapses in long-range connections and the more local connectivity of interneurons. This is consistent with recent experimental studies [
In this study, we have focused on the resting state of the network. However, it may also be used as a foundation for more functional investigations of cognition. For instance, the clusters of increased synaptic connectivity may serve as building blocks for cell assemblies [
In summary, we have provided a prefrontal cortex network model here with single cells and synapses strictly parametrized through in vitro electrophysiological findings (no specific tuning or adjustment of synaptic currents to compensate for simulated network size), with realistic cellular and synaptic heterogeneity, and with a structural layout derived from anatomical data. We have then systematically compared the full network activity to a number of spiking and correlation statistics from
We had shown previously that this model, although estimated from f-I and I-V curves only, can predict spike times under
We estimated neuron models for a large number of
Neurons were randomly connected with distinct connection probabilities
Apart from the recurrent synaptic connections within the network, we also introduced
Synapses were also equipped with short-term plasticity dynamics implemented by the corrected version [
parameter | U | ||
---|---|---|---|
E1 | 0.28 (0.02) | 194 (18) | 507 (37) |
E2 | 0.25 (0.02) | 671 (17) | 17 (5) |
E3 | 0.29 (0.03) | 329 (53) | 326 (66) |
I1 | 0.16 (0.10) | 45 (21) | 376 (253) |
I2 | 0.25 (0.13) | 706 (405) | 21 (9) |
I3 | 0.32 (0.14) | 144 (80) | 62 (31) |
Mean and standard deviation of the parameters of the six types of short-term synaptic plasticity.
Distributions of peak conductances (“synaptic weights”)
In a second step, since by far most of the studies cited above have been performed in sensory areas, data from laser scanning photostimulation (LSPS) [
The constant background currents
Two
Spike trains, voltage traces and local field potentials from the network simulation and the
Similarity among simulated and experimentally obtained distributions was assessed by two-sample Kolmogorov-Smirnov (KS) tests, where test statistic
Finally, we visualize the statistical overlap of distributions by plotting shaded areas representing the SEM around the mean at each value of the experimental distributions, which are computed from the 100 bootstrap samples as indicated above.
We are very grateful to Dr. Christopher Lapish, (Indiana University, Purdue University, Indianapolis) and Dr. Thomas Hahn (Central Institute of Mental Health, Medical Faculty Mannheim of Heidelberg University) for providing physiological recordings from rodent PFC for comparison with the model.