Autoregressive point-processes as latent state-space models

In 2016 I started a postdoc with the labs of Guido Sanguinetti and Matthias H. Hennig—and this is the first paper to result!

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A central challenge in neuroscience is understanding how the activity of single cells combines to create the collective dynamics that underlie perception, cognition, and behavior. One way to study this is to build detailed models, and "coarse grain" them to see which details are important. 

Our paper develops ways to relate detailed point-process models to coarse-grained quantities, like average neuronal firing rates and correlations. Point-process models are used for statistical modelling of spike train data. They can reveal effective neural dynamics by capturing how neurons in a population inhibit or excite each-other and themselves. 

Preview of figures: 

Figure 2: Moment closure of autoregressive PPGLMs combines aspects of three modeling approaches:

(A) Log-linear autoregressive PPGLM framework (e.g., Weber & Pillow, 2017). Dependence on the history of both extrinsic covariates x(t) and the process itself y(t) are mediated by linear filters, which are combined to predict the instantaneous log intensity of the process. (B) Latent state-space models learn a hidden dynamical system, which can be driven by both extrinsic covariates and spiking outputs. Such models are often fit using expectation- maximization, and the learned dynamics are descriptive. (C) Moment closure recasts autoregressive PPGLMs as state-space models. History dependence of the process is subsumed into the state-space dynamics, but the latent states retain a physical interpretation as moments of the process history (dashed arrow). (D) Compare to neural mass and neural field models, which define dynamics on a state space with a physical interpretation as moments of neural population activity

More colors for Matplotlib

 

The default Matplotlib color scheme is not good. While searching for something better for my thesis, I stumbled upon Bridget Riley's painting "Gather" in the RISD museum. Is uses five colors: white, black, ochre, blue, turquoise, and a rust-colored red. But, from a distance they combine to create something far more vibrant. The blue and the turquoise hues are similar, so perhaps not the best to combine in a plot, but the others form an attractive and distinct pallet:

White	#f1f0e9
Black	#44525c
Rust	#eb7a59
Ochre	#eea300
Azure	#5aa0df
Turquoise	#00bac9

Surprisingly, when I ran this through a color-blindness simulator , some colors become even more distinct, so it is fairly color-blind friendly. I added colors to expand the pallet, to handle gradients and plots with extra lines.

Yellow	#efcd2b
Moss	#77ae64
Mauve	#b56ab6
Magenta	#cc79a7
Violet	#8d5ccd
Indigo	#606ec3
Viridian	#11be8d
Chartreuse	#b59f1a

These are less distinct, and not all combinations are color-blind friendly. Here's a bit of Python to define and tests these colors:

from pylab import *
from matplotlib import patches
WHITE      = mpl.colors.to_rgb('#f1f0e9')
RUST       = mpl.colors.to_rgb('#eb7a59')
OCHRE      = mpl.colors.to_rgb('#eea300')
AZURE      = mpl.colors.to_rgb('#5aa0df')
TURQUOISE  = mpl.colors.to_rgb('#00bac9')
BLACK      = mpl.colors.to_rgb('#44525c')
YELLOW     = mpl.colors.to_rgb('#efcd2b')
INDIGO     = mpl.colors.to_rgb('#606ec3')
VIOLET     = mpl.colors.to_rgb('#8d5ccd')
MAUVE      = mpl.colors.to_rgb('#b56ab6')
MAGENTA    = mpl.colors.to_rgb('#cc79a7')
CHARTREUSE = mpl.colors.to_rgb('#b59f1a')
MOSS       = mpl.colors.to_rgb('#77ae64')
VIRIDIAN   = mpl.colors.to_rgb('#11be8d')
GATHER  = [WHITE,RUST,OCHRE,AZURE,TURQUOISE,BLACK]
COLORS  = [BLACK,WHITE,YELLOW,OCHRE,CHARTREUSE,MOSS,VIRIDIAN,TURQUOISE,AZURE,INDIGO,VIOLET,MAUVE,MAGENTA,RUST]
CYCLE   = [BLACK,RUST,AZURE,OCHRE,TURQUOISE,MAUVE,YELLOW,INDIGO]
mpl.rcParams['axes.prop_cycle'] = mpl.cycler(color=CYCLE)
def test_panel(COLORS):
    ax = gca()
    NCOLORS = len(COLORS)
    axis('off')
    xlim(0,NCOLORS)
    ylim(NCOLORS,0)
    for j in range(NCOLORS):
        for i in range(NCOLORS):
            ax.add_patch(patches.Rectangle((i,j),1,1,linewidth=1,
                edgecolor='none',facecolor=COLORS[i]))
            ax.add_patch(patches.Rectangle((i+.25,j+.25),.5,.5,linewidth=1,
                edgecolor='none',facecolor=COLORS[j]))
    axis('equal')
figure(figsize=(6,3),dpi=200)
subplots_adjust(0,0,1,1,0.1,0)
subplot(111)
test_panel(COLORS)
savefig('colorblind_test_panel.png',bbox_inches='tight',pad_inches=0)
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Two squences form decent color maps. They aren't perceptually uniform, but pass the color blindness simulator tests. Adding violet, mauve, and moss, to the original Gather pallet creates a hue wheel. This one flunks color-blindness test (see Crameri et al. ). Here they are in python:

riley2 = matplotlib.colors.LinearSegmentedColormap.from_list('riley2',
    [INDIGO,VIOLET,MAUVE,MAGENTA,RUST,OCHRE])
riley3 = matplotlib.colors.LinearSegmentedColormap.from_list('riley3',
   [OCHRE,CHARTREUSE,MOSS,VIRIDIAN,TURQUOISE,AZURE])
# Smoothed out mauve, violet, rust, ochre, moss, turquoise, azure, in a loop
hues = ['#8c62cc', '#9560c8', '#9e62c3', '#a765be', '#b068b6', '#bb6caa', '#c66f98',
        '#d27384', '#dd7670', '#e67c5c', '#ea8348', '#ec8b34', '#ec9421', '#e99c13',
        '#dea212', '#c9a620', '#b0a934', '#96ab4a', '#7cae60', '#62b076', '#48b38c',
        '#2eb5a2', '#1ab7b6', '#12b6c5', '#1cb2ce', '#2dadd4', '#41a7d8', '#529fdb',
        '#6194db', '#6d86d8', '#7977d4', '#836ad0']
riley = matplotlib.colors.LinearSegmentedColormap.from_list('riley',hues)
# Make new maps behave like native Matplotlib maps
plt.register_cmap(name='riley2',cmap=riley2)
plt.register_cmap(name='riley3',cmap=riley3)
plt.register_cmap(name='riley' ,cmap=riley )
# Show as figure
figure(figsize=(5,1),dpi=300)
subplot(311)
imshow([linspace(0,1,256)],cmap='riley2',aspect='auto')
axis('off'); tight_layout()
subplot(312)
imshow([linspace(0,1,256)],cmap='riley3',aspect='auto')
axis('off'); tight_layout()
subplot(313)
imshow([linspace(0,1,256)],cmap='riley',aspect='auto')
axis('off'); tight_layout()
subplots_adjust(hspace=0.3)
savefig('moremaps.png')
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