Simulating lesions in multi-layer, multi-columnar model of neocortex

Abstract

The paper presents results of modeling global and focal loss of layers in a multi-columnar model of neocortex. Specifically, the spread of activity across columns in conditions of inhibitory blockade is compared. With very low inhibition activity spreads through all layers, however, deep layers are critical for spread of activity when inhibition is only moderately blocked.

I. Introduction

The laminar organization of the neocortex is vital to its normal operation. The neocortex consists of several layers that differ in thickness, number of neurons and their types, as well as types of input they receive, and the ways that input is processed. Analyzing flows of activity in vertical and horizontal brain tissue slices have been a significant source of knowledge about neuronal connections in the brain. In particular, there is a wide range of studies that compare different aspects of connectivity between superficial and deep layers [1], [2], [3].

Although such analysis can provide valuable insight into brain’s network dynamics and validation of computational models, focal or global loss of layers has not been modeled using computational models, even in those that preserve laminar structure of cortex [4].

In this paper we focus on analyzing the spread of activity with different levels of inhibitory blockade in superficial and deep layers in a multi-layer, multi-columnar model of neocortex. The model used, a network of spiking neurons, employs two inhibitory and two excitatory neuronal types, detailed neural connections both within and between the layers and columns, and a synapse model that allows for modeling short-time plasticity. Parameters of the connections, such as their probabilities, maximal amplitude, half-width of post-synaptic potential (PSP), and latency to peak of PSP differ according to types and location of the connected neurons.

There are two main approaches to modeling [5]. In the first approach, single neurons are modeled and then interconnected into larger networks. The second approach, referred to as “lumped”, focuses on macroscopic interactions between neuronal populations. The presented model belongs to the first group but incorporates several aspects of neuronal connectivity instead of employing a detailed compartmental model of a neuron.

II. Methods

A. Network model

The spatial structure of the artificial neural network model is consistent with rat somatosensory cortex with four layers: II/III, IV, V, and VI. The simple neuron model developed by Izhikevich [6] and the short-term plasticity model developed by Tsodyks and Markram [7], [8] are used in the network.

The neuron model is a two dimensional system of nonlinear ordinary equations of the form

$$\begin{gathered} v^{\prime }=0.004v^2+5v+140-u+I \\ {u}^{\prime }=a\left(bv-u\right) \end{gathered}$$ (1)

with the condition

$$\text{if}\hspace{0.33em}v\ge 30,\hspace{0.33em}\text{then}\left\{\begin{array}{ll}v\hspace{1em}\leftarrow \hspace{1em}\hfill & c\hfill \\ u\hspace{1em}\leftarrow \hspace{1em}\hfill & u+d,\hfill \end{array}\right.$$ (2)

where v represents the membrane potential of the neuron and u is a membrane recovery variable (both are functions of time), a, b, c and d are dimensionless parameters and I is the value of the input to the neuron. The membrane potential v has mV scale and the time has ms scale. Depending on the values of parameters in (1) and (2) this model can mimic spike pattern of different types of neurons. In order to restrict maximal firing rates of neurons, a modified version of the model is used. Generation of an action potential is prevented if the time from the previous spike is shorter than given by the maximum firing frequency for the particular neuronal subtype.

The network uses four neuron types. Two are excitatory, the regular spiking (RS) and intrinsically bursting (IB), and two are inhibitory, the fast-spiking (FS) and low-threshold spiking (LTS) neurons, which are generated using the same parameters as in [6]. Spatial distribution of the four neuron types within layers is shown in Figure 1. The total number of neurons used, arranged into five cortical columns, is 3940, equivalent to 5% of the number of cells found in rat somatosensory cortex.

Fig. 1.

Example of connectivity. (a) RS neurons (light green triangle) in layer VI connect with different probabilities to IB neurons in layer V (blue-green triangle), FS neurons (blue circle) in layer VI within the same and adjacent columns, LTS cells (red ellipse) in layer VI, and other RS cells in layers V and VI. For each layer, the number in parenthesis is the total number of neurons in this layer per column. Percentages of each neuron type within given layer are also listed. (b) Selected cells (as shown) within each column receive input from a thalamic cell.

Short-time plasticity is an inherent dynamic of synapses resulting in different responses of post-synaptic neurons for different temporal patterns of pre-synaptic spikes [9]. Specifically, the post-synaptic response can be smaller (depression) or larger (facilitation) than the previous one [10] This mechanism is known to be crucial for modeling and understanding the synchronous behavior of the network and various learning mechanisms [11].

The model of short-time dynamic consists of four equations [12]

$$\left\{\begin{array}{l}{x}^{\prime }=\frac{z}{{\tau }_{rec}}-ux\delta \left(t-t_{pres}\right)\\ y^{\prime }=-\frac{y}{{\tau }_I}+ux\delta \left(t-t_{pres}\right)\\ z^{\prime }=\frac{y}{{\tau }_I}-\frac{z}{{\tau }_{rec}}\\ u^{\prime }=-\frac{u}{{\tau }_{fac}}+U\left(1-u\right)\delta \left(t-t_{pres}\right)\end{array}\right.$$ (3)

Here x, y, and z are fractions of synaptic resources in the recovered, active and inactive states respectively, t_pres is the time of the pre-synaptic spike, τ_I is the decay constant of the post-synaptic current, and τ_rec represents the recovery from the synaptic depression. The variable u is the fraction of the available resources used by the pre-synaptic spike. It increases with each pre-synaptic spike (this change is described by constant U ) and decays accordingly to τ_fac.

We gathered data about connections between different neuron types from studies of paired intracellular recordings [13], [14], [16], [17], [18], [19]. Connections are described using the probability of connection (see Figure 1), the average strength of connection, the type and rate of short-term plasticity (STP), and the shape of post-synaptic potential (PSP), i.e., its time to peak and half-width. These characteristics of the connections reflect the spatial structure of the network, e.g., the distribution of the number of synaptic connections and their location on the dendritic tree. Incorporating all these parameters into the network is crucial for modeling activity of the network and for compensating for not modeling various compartments of a neuron.

The thalamic input is modeled as a single cell that is connected to select cells within a single column: RS and FS neurons in layer IV and RS neurons in layer V. This arrangement is consistent with the processes that take place in rat somatosensory cortex, where thalamic afferents synapse in layer IV, including onto the apical dendrites of layer V pyramidal neurons as those dendrites pass through layer IV. This kind of input is used when the stimulated column is not lesioned. In simulations of only a single layer we used localized input, that is a current provided to a group of RS neurons within the stimulated column. Gaussian noise is added to all neurons in order to simulate their embedding in a larger network (the brain).

Local field potentials (LFP) are computed by adding the voltage of excitatory pyramidal cells in layers III and V, within a single column. The sign of the calculated sum was reversed to make it consistent with the measured LFP which is biologically measured from outside the cells.

The network has been shown to exhibit a range of biologically accurate behavior, including columnar and laminar flow of focal sensory input, balance between vertical and horizontal inhibition, generation of gamma oscillations, and ability to produce different stages of epileptiform activity with different levels of inhibitory blockade [15].

B. Simulations

The simulations were performed using time step of 0.1 ms on a network consisting of five columns with added white noise with variance of eight. Inhibitory blockade was modeled by decreasing the weights of all inhibitory connections. The amplitude of the stimulus was 8 mV, however, simulations with stimulus amplitude in the range 7 – 19 mV were performed and the results were consistent.

III. Results

A. Spread of activity in the intact network

First, different levels of inhibitory blockade within the whole network where simulated. As expected, without inhibitory blockade, or with low levels of blockade, there is little spread to adjacent columns in response to stimulation within a single column. This can be seen from the peak negativity (excitation) of the LFP plotted as a function of the level of inhibitory blockade (Figure 2). However, with 40% blockade, activity in the adjacent column is near that within the stimulated column and continues to propagate horizontally across the columns. This is a validation of the well known function of inhibition in limiting horizontal spread within neocortex [20].

Fig. 2.

Spread of activity in the intact network. (a) Peak negativity vs. level of inhibitory blockade in different columns. Results averaged from two simulations. (b) Local field potentials in stimulated column (column 2), one column away (column 3), and two columns away (column 4) in conditions of 70% inhibitory blockade.

B. Global horizontal cuts

Second, all layers but one were removed across all columns and the remaining layer ‘strip’ was stimulated focally in the second column. Different levels of inhibitory blockade were applied.

Propagation in a strip of layer III (Fig. 3a,c) and a strip of layer V (Fig. 3b,d) was investigated. Comparing Figures 3a and b with Figure 2a we notice that the activity spreads faster with the increase of the inhibitory blockade level in the intact network (with all layers) than in a single layer. In the whole network with 40% blockade, the activity in adjacent columns is comparable to the stimulated column, whereas in network consisting of only one layer these levels of activity do not become similar until the blockade reaches 80–90%. In addition, LFPs in the lesioned network are shorter and of lower amplitude when compared with the intact network.

Fig. 3.

Horizontal spread of activity through individual layers after removal of other layers in all columns (global lesion). (a-b) Peak evoked negativity from computed LFP after stimulation of column 2 under various levels of inhibitory blockade. Cortical strips of only layer III (a) or layer V (b) were modeled as created experimentally from biological tissue after cut of coronal slices (Telfeian). Average of 2 simulations shown. (c-d) Local field potentials produced after stimulation of column 2 under condition of 70% inhibitory blockade for layer III (c, black) and layer V (d, blue). Activity in columns 3 and 4 is a result of propagation across the laminar strip. Propagation to columns 3 and 4 occurs for both layer III and layer V strips.

At high levels of inhibitory blockade (80–90%), activity propagates across columns for both the layer III and layer V strip. Importantly, in the case of 70% blockade some spreading is noticeable within layer V, but not within layer III. This result is consistent with what was reported for the biological network in [1].

C. Focal loss of layers

Next, either superficial (III and IV) or deep (V and VI) layers were removed within column 3 only, leaving remaining layers to bridge columns 2 and 4. To determine if activity could spread across the bridge, stimulation was applied focally within column 2.

Little propagation is observed with 20–40% blockade, but with 60% blockade, the spread is strong for the deep but not the superficial layer bridge (Figure 4a,b). Moreover, LFPs show that for the deep layer bridge, the activity in column number four (two columns away from the stimulation) is comparable to that in the intact network (see Figure 2b). This supports the idea that layer V may be more susceptible to activity propagation when all columns are intact. It has been suggested that this is true biologically due to IB cells that are known to have horizontal projections [21] and to be more active during epileptiform activity relative to the RS cells [20]. Within our model, we also find that with 40% inhibitory blockade and all layers intact, the average RS firing rate is 12 spikes/sec, while that of IB cells is 26.

Fig. 4.

Propagation through superficial versus deep cortex measured by creating bridges of tissue within column 3 (focal lesions). (a-b) Peak evoked negativity from computed LFP after stimulation of column 2 under various levels of inhibitory blockade when only superficial (a) or deep (b) layers remain within column 3. Average of 2 simulations shown. Larger LFPs are produced from propagation across the deep layer bridge. (c-d) Local field potentials produced after stimulation of column 2 under condition of 60% inhibitory blockade for the superficial (c, black) and deep(d, blue) layer bridge. A greater amount of propagation occurs with the deep layer bridge.

Finally, as was true for the global lesion, with high levels of inhibitory blockade, both superficial and deep layer bridges can support spread and propagation of excitatory activity across columns (Figure 4a,b).

IV. Conclusions

Here we have examined the propagation of activity across the computational multi-layer, multi-columnar cortex. As expected and previously shown, with all layers and inhibition intact, stimulation within one column remains focal. As shown biologically, and here shown within our computational model, even low level blockade of inhibition will allow the horizontal propagation of activity across columns [20], [21]. We show further here that deep layers are distinct from superficial layers in this ability. While both superficial and deep layers can support the propagation of activity at high levels of inhibitory blockade, the threshold at which the propagation succeeds is lower for deep layers. In addition, the deep layer response looks most similar to that produced when all layers are intact, further suggesting that the deep layers are normally a significant pathway for propagation under conditions of reduced inhibition.

All of these results are consistent with biological findings from cortical slices under similar conditions of creating strips and laminar bridges [1], [2]. Importantly, this model is the first to use multiple layers, multiple columns and short term plasticity for the computational study of propagating activity.

The results suggest that this model can be used to accurately model epileptiform activity, as well as changes to the laminar structure. In particular, it can be used for modeling neuronal abnormalities